The GROMOS Software for Biomolecular Simulation:...

The GROMOS Software for Biomolecular Simulation:GROMOS05

MARKUS CHRISTEN, PHILIPPE H. HUNENBERGER, DIRK BAKOWIES, RICCARDO BARON,ROLAND BURGI,* DAAN P. GEERKE, TIM N. HEINZ, MIKA A. KASTENHOLZ,

VINCENT KRAUTLER, CHRIS OOSTENBRINK,† CHRISTINE PETER,‡ DANIEL TRZESNIAK,WILFRED F. VAN GUNSTEREN

Laboratory of Physical Chemistry, Swiss Federal Institute of Technology Zurich,ETH-Honggerberg, CH-8093 Zurich, Switzerland

Received 22 July 2005; Accepted 2 August 2005DOI 10.1002/jcc.20303

Published online in Wiley InterScience (www.interscience.wiley.com).

Abstract: We present the latest version of the Groningen Molecular Simulation program package, GROMOS05. It hasbeen developed for the dynamical modelling of (bio)molecules using the methods of molecular dynamics, stochasticdynamics, and energy minimization. An overview of GROMOS05 is given, highlighting features not present in the lastmajor release, GROMOS96. The organization of the program package is outlined and the included analysis packageGROMOS�� is described. Finally, some applications illustrating the various available functionalities are presented.

© 2005 Wiley Periodicals, Inc. J Comput Chem 26: 1719–1751, 2005

Key words: molecular dynamics simulation; programming; GROMOS; biomolecular simulation

Introduction

Starting with GROMOS80, the GROMOS program package hasbeen developed over the past 25 years to facilitate research effortsin the field of biomolecular simulation in a university environment.The GROMOS software was and is meant for use in a scientificenvironment, which may be characterized by a continuouslychanging flow of users, who either wish to investigate and imple-ment new simulation algorithms or intend to carry out applicationsof simulation in a variety of fields, ranging from polymers, glassesand liquid crystals, to crystals and solutions of biomolecules (pro-teins, nucleic acids, saccharides, and lipids). To this purpose,GROMOS has been developed based on the following principles:(1) transparency of the code, making modifications easy; (2) mod-ular architecture, so that only parts of it need be modified for theimplementation of new functionalities designed by users; (3) in-dependence of the simulation code and the force field; and (4)independence of the simulation code and the computer hardware.The major releases of the GROMOS software are GROMOS871,2

developed at the University of Groningen, GROMOS963,4 devel-oped at ETH Zurich, and now GROMOS05. GROMOS has foundwidespread use (hundreds of licenses in over 57 countries on allcontinents except Antarctica, see Fig. 1), triggered by the fact thatit has been designed for ease of extendability and that the completesource code is made available to research establishments for anominal fee (http://www.igc.ethz.ch/gromos). The program code

has been further developed in the group for computational chem-istry at ETH Zurich (Switzerland) throughout the recent years,leading now to a new major release, GROMOS05. The enhance-ments were governed by the following criteria: (1) interest of ourresearch group,5 (2) ease of use, (3) extendability, (4) demon-strated usefulness or efficiency of new methods, (5) well-definedand correct formulae and algorithms, and (6) computational effi-ciency. The second criterium led to a complete rewrite of the setupand analysis tools, now contained in the GROMOS�� setup andanalysis subpackage, written in C��. The third criterium led to arewrite in C�� of the MD engine, the part that carries outmolecular dynamics (MD) or stochastic dynamics (SD) simula-tions as well as energy minimizations (EM), into a new programcalled MD��. In parallel, the original FORTRAN version of the

Correspondence to: W. F. van Gunsteren; e-mail: [email protected]

Contract/grant sponsor: National Center of Competence in Research(NCCR) Structural Biology of the Swiss National Science Foundation(SNSF)

*Present address: Swissre, Zurich, Switzerland.†Present address: Vrije Universiteit, PharmaceuticalSciences/Pharmacochemistry, De Boelelaan 1083 P262, NL-1081 HVAmsterdam, The Netherlands.‡Present address: Max-Planck-Institute for Polymer Research,Ackermannweg 10, D-55128 Mainz, Germany.

© 2005 Wiley Periodicals, Inc.

MD engine, PROMD, was further developed to introduce manynew features (some of which are not yet available in MD��). Inthe long term (beyond GROMOS05), MD�� will entirely replacePROMD.

In the next sections the main features of GROMOS05 aredescribed. First, new functionalities with respect to GROMOS96are highlighted. Second follows the algorithmic description ofselected new functionalities. Third, the organization of the code isdiscussed and an overview of the programs present in the GRO-MOS�� analysis subpackage is provided. Fourth, examples ofapplications are reported for some of the newer features. Finally,summary and conclusions are provided.

Overview of Functionalities

Here, the main features of the two MD engines available inGROMOS05, PROMD and MD�� are described. These twoprograms share most of the basic functionalities, but still differ ina number of aspects. The FORTRAN MD engine (PROMD)retains all features of the GROMOS96 release and adds a numberof new functionalities. The C�� MD engine (MD��) containsmost of the GROMOS96 features (except four-dimensional andpath-integral simulations), a subset of the new functionalities re-cently introduced into PROMD (since the GROMOS96 release),and some new features of its own.

A nonexhaustive list of the features included is:

● molecular dynamics (MD) simulation, stochastic dynamics (SD)simulation, and energy minimization (EM; steepest descent orconjugate gradient);

● periodic boundary conditions (vacuum, rectangular, truncated

Figure 1. Distribution of GROMOS licenses.

Figure 2. Illustration of the standard double-loop and the improvedpairlist algorithm for a set of 10 atoms or charge groups. The standardalgorithm scans the triangular atom-pair matrix row by row (top,unshaded box). A quadratic (top, light gray shade) or rhombic (top,dark gray shade) window scans more atom pairs for a given number ofatoms loaded into processor cache. The triangular atom-pair matrixmay be reordered to become rectangular (bottom), in which case therhombic window becomes quadratic.

1720 Christen et al. • Vol. 26, No. 16 • Journal of Computational Chemistry

octahedral, or triclinic computational box; possibility of per-forming multiple-unit-cell simulations);

● temperature control (constraining, weak coupling, Nose–Hooveror Nose–Hoover chain; possible coupling of different subsets ofdegrees of freedom to separate temperature baths);

● pressure control (weak coupling or Andersen–Parrinello–Rah-man, isotropic, partially anisotropic, and fully anisotropic coor-dinate scaling; atom-based or group-based pressure definition);

● long-range electrostatic interactions: straight cutoff truncation,truncation with Poisson–Boltzmann reaction field (RF) correc-tion and lattice-sum (LS) methods, including Ewald summationand particle–particle–particle–mesh (P3M);

● charge-group based or atom-based cutoff for the nonbondedinteractions;

● Grid-based pairlist construction;● nonphysical interactions: atom-position, atom-distance, dihe-

dral-angle, NOE, and J-value restraints as well as atom-positionand atom-distance constraints (SHAKE, M-SHAKE, LINCS);

● enhanced sampling: local elevation MD, replica exchange MD(REMD), and umbrella sampling;

● calculation of free energy changes based on the coupling pa-rameter (�) approach using thermodynamic integration, slow-growth or one-step perturbation, possibly including soft-corenonbonded interactions;

● path-integral simulation;● MPI and OMP parallelization.

A number of the new features introduced in GROMOS05 arediscussed later. Preexisting features have been described in detailselsewhere.3,4 The functionalities of the pre- and postprocessingprograms contained in GROMOS�� are also discussed later. Acomplete description of the available features will be included inthe new GROMOS manuals.

Algorithms

MD Algorithm

The complete MD algorithm based on the leap-frog scheme asimplemented in GROMOS is the following.3 Given initial atomicpositions and velocities, which satisfy any given geometrical con-straints:

1. Save positions (reset atomic coordinates into the referencecomputational box in case of periodic boundary conditions)and velocities for later analysis.

2. Remove center of mass motion (if required).3. Calculate (unconstrained) energies, forces, and virial contri-

bution from the potential energy function (using the nearestimage convention in case of periodic boundary conditions).Save these.

4. Enforce any given position constraints by resetting the forcesand velocities of positionally constrained atoms to zero.

5. Update the velocities using the leapfrog scheme.6. Apply temperature coupling (constraining, weak coupling,

Nose–Hoover or Nose–Hoover chain) by scaling the atomvelocities.

7. Update the positions using the leapfrog scheme.8. Enforce distance constraints (using SHAKE, M-SHAKE, or

LINCS) both for positions and velocities, and calculate thecorresponding forces and virial contribution. Save these.

9. Calculate the kinetic energy and temperature (possibly on thebasis of separate subsets of degrees of freedom).

10. Calculate the pressure (atom-based or group-based pressuredefinition).

11. Apply pressure scaling (weak coupling or Andersen–Parrinel-lo–Rahman) by scaling atomic positions (isotropic, partiallyanisotropic, or fully anisotropic scaling).

12. Update the coupling parameter � for (free energy) simulationsinvolving � changes (slow growth).

13. Calculate total energies, averages, and fluctuations. Savethese.

This sequence is repeated for the required number of simulationsteps.

New Features

Spatial Boundary Conditions

Spatial boundary conditions are defined by the shape, size, andorientation of the simulated system, and the nature of the boundaryto its surroundings. The GROMOS05 implementation (bothPROMD and MD��) admits four types of boundary conditions:(1) vacuum boundary conditions; (2) periodic boundary conditionsbased on a rectangular box; (3) periodic boundary conditions basedon a truncated-octahedral box; (4) periodic boundary conditionsbased on a triclinic box. In the three latter cases, the system isconfined to a (reference) computational box that is surrounded byan infinite number of periodic copies of itself.

When periodic boundary conditions are applied, the shape, size,and orientation of the computational box must be defined. Forrectangular and triclinic periodic boundary conditions, this is doneby specifying the three edge vectors a, b, and c (defining aright-handed coordinate system) of the reference computationalbox. For a truncated-octahedral box, these vectors correspondinstead to the edges of the cube based on which the truncatedoctahedron is constructed. In practice, the three vectors are spec-ified by their lengths a, b, and c, the box angles � (between a andb), � (between a and c), and � (between b and c) they defineamong each other (all in the range ]0; �[), and the three Eulerrotation angles �, �, and � (the two former ones in the range ]��;�], the latter one in the range [��/2; �/2]) characterizing theorientation of the box relative to the reference right-handed Car-tesian coordinate system (ex, ey, ez). To define the Euler angles,the three edge vectors are used to define a box-linked right-handedCartesian coordinate system (ex�, ey�, ez�) in the following way:(1) the x�-axis is chosen along and in the direction of a; (2) they�-axis is chosen orthogonal to a in the plane defined by a and b,and oriented in the direction of b; (3) the z�-axis is chosenorthogonal to both a and b, and oriented in the direction of c. Thereference coordinate system can be rotated onto the box-linkedcoordinate system by the following series of rotations: (1) arotation by an angle � around the z-axis; (2) a rotation by an angle� around the new y-axis; (3) a rotation by an angle � around the

GROMOS Software for Biomolecular Simulation 1721

new x-axis. The angles �, �, and � thus represent the three Eulerrotation angles in a zyx or yaw-pitch-roll convention. The use of arectangular or truncated-octahedral box requires � � � � � � �/2and is restricted to nonrotated boxes with � � � � � � 0. The useof a truncated-octahedral box also requires a � b � c. In the caseof vacuum boundary conditions, the system is nonperiodic, and a,b, and c need not be specified.

Based on a general triclinic box in an arbitrary orientation, theposition of an atom may be specified through coordinates r � ( x,y, z) within the reference Cartesian coordinate system (ex, ey, ez),or through oblique fractional coordinates � � (u, v, w) withreference to the box-edge vectors. The two types of coordinates arerelated by

r L� �, (1)

where the matrix L� contains the components of a, b, and c in thereference Cartesian coordinate system as its columns. The boxvolume is

V �L� �. (2)

This matrix can be decomposed as

L� �ax bx cx

ay by cy

az bz cz

� R� S� , (3)

where the orthogonal transformation matrix R� (rotation betweenreference and box-linked Cartesian coordinate systems) is given by

R� �cos � cos � sin � sin � cos � cos � sin � cos � sin � cos � � sin � sin �cos � sin � sin � sin � sin � � cos � cos � cos � sin � sin � sin � cos �

�sin � sin � cos � cos � cos ��, (4)

and the transformation matrix S� (between box-linked Cartesiancoordinates and oblique fractional coordinates) is given by

S� �a b cos � c cos �0 b sin � c sin � cos �0 0 c sin � sin �

�, (5)

with

cos � cos � cos � cos �

sin � sin �, � � ]0; ��. (6)

As shown by Bekker,6 a simulation performed in a truncated-octahedral box can equivalently be performed in a special type oftriclinic box, by applying an appropriate coordinate transforma-tion. A possible choice for the edges at, bt, and ct of the trans-formed triclinic box is

at a, bt �1/2��a � b � c� and ct �1/2��a b � c�.

(7)

The corresponding box-edge lengths, box angles, and Euler anglesare at � a, bt � ct � (�3/ 2)a, �t � acos(�1/3) 109.5o,�t � acos(�1/�3) 125.3o, �t � acos(1/�3) 54.8o, �t ��t � 0, and �t � 45o. The mapping of atomic coordinates withina truncated-octahedral box to atomic coordinates within the trans-formed triclinic box is performed by applying shifts along the at,bt, and ct vectors. This formalism is applied for the generalizationof grid-based pairlist algorithms and lattice-sum electrostatics totruncated-octahedral boxes. Because the truncated-octahedral casecan always be mapped to the triclinic case, subsequent sectionswill only discuss the case of the triclinic box.

Multiple-Unit-Cell Simulations

Within the GROMOS05 implementation (both PROMD andMD��), it is possible to simulate a periodic computational box(rectangular or triclinic only) consisting of multiple periodic cop-ies of a smaller unit cell (referred to here as subcells). This optionmay be useful when trying to simulate a single unit cell of a crystalthat is too small to allow for the application of a reasonably largecutoff value. The number of subcell boundaries along the threebox-edge vectors a, b, and c are Ma, Mb, and Mc, so that the totalnumber of subcells is M � Ma � Mb � Mc. In this case, the totalnumber of solute molecules and the total number of solvent mol-ecules must both be integer multiples of M.

Subcell periodicity within the computational box must be ful-filled by the initial coordinates and velocities, and will then bemaintained throughout the simulation. The corresponding period-icity constraints on the forces, velocities, and coordinates arechecked at each step of the simulation, with reference to the subsetof solute and solvent molecules with the smallest sequence num-bers. Deviations smaller than user-specified tolerances (numericaldrift) are systematically corrected. Deviations larger than the tol-erance are reported as an error.

Note that the removal of the center of mass motion, wheneverrequired, is applied to charge groups and solvent molecules gath-ered in the individual subcells. Note also that the application of theparticle–particle–particle–mesh (P3M) method to evaluate electro-static interactions will only give rise to exactly periodic forces ifthe number of P3M grid subdivisions along each axis is an integermultiple of the corresponding number of subcell boundaries.

In MD��, only a single subcell is simulated. Just for thenonbonded interaction calculation the subcell is multiplied toconstruct the full box. Energies, forces, and virial contributionsneed only be calculated for atoms inside the reference subcell, but


those are interacting with all other atoms in the full box. Becauseof that, fewer (nonbonded and covalent) interactions than in thefull box simulation have to be calculated and the positions andvelocities are always exactly periodic.

Rigid-Body Motion

The laws of classical mechanics lead to two conserved quantities(besides the total energy): (1) the linear momentum psys of thesystem, and (2) the angular momentum Lsys of the system aroundits center of mass. In simulations under periodic boundary condi-tions, the two quantities refer to the infinite periodic system.However, in this case, if the linear momentum pbox of the com-putational box is also conserved, the corresponding angular mo-mentum Lbox is not (because correlated rotational motions in twoadjacent boxes exert friction on each other, leading to an exchangeof kinetic energy with the other degrees of freedom of the system).Furthermore, the quantity Lsys must vanish (because overall uni-form rotation of the infinite periodic system would lead to nonpe-riodic centrifugal forces). When SD is applied instead of MD, thepresence of random and frictional forces couples the system (orbox) linear and angular momenta with the other degrees of free-dom of the system, so that these quantities are no longer con-served. The inclusion of special (unphysical) forces, such as atom-position restraining or constraining forces on a subset of atoms inthe system, may also lead to nonconservation of these quantities.The above observations are summarized in Table 1.

The physical properties of a molecular system are independentof psys (or pbox). However, for MD simulations under vacuumboundary conditions, they depend on Lsys, because the rotation ofthe system leads to centrifugal forces. For these reasons, in theGROMOS05 implementation (PROMD only), the constraintpsys � 0 (or pbox � 0) is imposed at each time step throughout anysimulation. In addition, the constraint Lsys � Lsys

o , where Lsyso is

a user-specified reference value, is imposed throughout any MDsimulation under vacuum boundary conditions. These two con-straints will in particular prevent the progressive accumulation ofkinetic energy into the uncoupled degrees of freedom due toapplying a thermostat by velocity scaling and numerical errors,giving rise to the well-known (and quite unpleasant) “flying icecube problem.”7–9 In addition roto-translational constraints10 maybe applied to the solute molecule(s) during the simulation.

Instantaneous Temperature and Pressure

The instantaneous observables � and �, and time averages ofwhich determine the system macroscopic temperature T and pres-sure tensor P� , are not uniquely defined.7,11–14 Acceptable alterna-tive definitions differ by any quantity with a vanishing equilibriumaverage. Note, however, that the corresponding equilibrium fluc-tuations depend on the specific definition chosen for the instanta-neous observable.

In the GROMOS05 implementation (both PROMD andMD��), the instantaneous temperature � is defined using the(atom-based) internal kinetic energy of the system, as7

� 2

kBNdf�, (8)

where kB is Boltzmann’s constant, Ndf the number of internal(unconstrained) degrees of freedom of the system, and � itsinstantaneous internal kinetic energy. The word “internal” is usedhere to exclude possible contributions from the degrees of freedomthat are “external,” that is, uncoupled from the system in terms ofkinetic energy exchange.15 In MD simulations, these are the de-grees of freedom associated with the system (or box) rigid-bodytranslation and, under vacuum boundary conditions, system rigid-body rotation. The number of internal degrees of freedom is thuscalculated as three times the total number N of atoms in thesystem, minus the number Nc of geometrical constraints, minus thenumber Nr of external degrees of freedom (Table 1), that is,

Ndf 3N Nc Nr. (9)

The instantaneous internal kinetic energy is defined as

� 1

2 �i�1

N

miri2, (10)

where the internal (also called peculiar) velocities ri are obtainedfrom the real atomic velocities ri

o by excluding any componentalong the external degrees of freedom (it is assumed that thevelocities ri

o are already exempt of any component along possiblegeometrical constraints). Due to the constraints imposed in the

Table 1. Properties of Momenta Associated with Rigid-Body Motions in MD or SD Simulations underVacuum or Periodic Boundary Conditions.

Method Boundary psys pbox Lsys Lbox Nr

MD Vacuum Conserved Conserved 6MD Periodic Infinite Conserved Zero Coupled 3SD Vacuum Coupled Coupled 0SD Periodic Infinite Coupled Zero Coupled 0

The quantities considered are: psys and pbox (linear momentum of the overall system and the computational box, Lsys

and Lbox (angular momentum of the overall system and the computational box), and Nr (number of uncoupled degreesof freedom associated with rigid-body motions).


GROMOS05 implementation (PROMD only) on the system totallinear and angular momenta, the internal velocities only differfrom the real ones when MD is applied under vacuum boundaryconditions. In this case, one has

ri rio I�CM

�1 �ro�Lsyso �ri

o rCMo �, (11)

where rCMo is the (constant) coordinate vector of the system center

of mass, Lsyso the (constant) system angular momentum about the

CM, and I�CM is the (configuration-dependent) inertia tensor of thesystem relative to the CM. The latter quantity is defined as

I�CM�ro� �i�1

N

mi�rio rCM

o � � �rio rCM

o �, (12)

where a R b denotes the tensor with elements �, � equal to a�b�.In the GROMOS05 implementation (both PROMD and

MD��), the instantaneous pressure tensor � is related to thegroup-based virial and group-based internal kinetic energy tensorof the system. The word “group-based” refers to a pressure defi-nition excluding virial and kinetic-energy contributions withinuser-specified groups of (covalently-linked) atoms.13,14 Thesegroups will be referred to as virial groups. Single atoms can beused as virial groups, in which case an atom-based pressure defi-nition is recovered. The average pressure is not affected by thespecific choice of groups, but the pressure fluctuations are. Inpractice, atom grouping is used to reduce these fluctuations. Thepressure is only calculated for systems under periodic boundaryconditions. Note also that the contribution of special (nonphysical)forces (e.g., atom-position or atom-distance restraining) to thepressure is not included.

The instantaneous atom-based pressure tensor is computed as

�* 2

��* �*� (13)

where

�* 1

2 �i�1

N

miri � ri, (14)

and

�*�� 1

2 ��

��

�L��

L�� (15)

are the instantaneous atom-based internal kinetic energy and virialtensors, � and � being the instantaneous volume and total poten-tial energy of the system, L� the matrix defined by eq. (3), and ri theinternal velocities introduced above. The corresponding isotropic(scalar) quantities are related to the tensor quantities through

�* Tr��*, �* Tr��* and �* �1/3�Tr��*, (16)

where Tr returns the trace of a matrix, �* is equivalent to � in eq.(10), and �* is defined as

�* 3�

2

��

��. (17)

It is possible to show that:16,17 (1) the contribution to theatom-based virial tensor of a potential energy term that solelydepends on the scalar products or determinants defined by a set ofinteratomic vectors is symmetric; (2) the contribution to the atom-based virial tensor of a potential energy term that solely dependson the angles defined by a set of vectors is (in addition) traceless.The first observation implies that all covalent (bond stretching orconstraint, bond-angle bending, proper and improper dihedral an-gle), and pairwise nonbonded force field terms lead to a symmetriccontribution to the atom-based virial. The second observationimplies that covalent bond-angle bending as well as proper andimproper dihedral angle (but not bond stretching or constraint andpairwise nonbonded) terms lead to a traceless contribution to theatom-based virial (i.e., no contribution to the scalar atom-basedpressure). However, these results are generally not valid for thecorresponding group-based tensor (see below).

In the special case of a pairwise-additive interaction term �p

depending on minimum-image interatomic distances and withoutexplicit dependence on the box dimensions (bond stretching orconstraint and pairwise nonbonded terms; but not reciprocal-spacelattice-sum interactions13,14), eq. (15) leads to a virial contribution

�*p �1

2 �i

N �j�i

N

Fp,ij � r� ij, (18)

where rij � ri � rj is the vector connecting j to i, r�ij thecorresponding minimum-image vector, and Fp,ij the pairwise forceexerted by atom j on atom i. This equation is easily generalized tointeraction terms involving more than two atoms (bond-anglebending, proper and improper dihedral-angle terms). The atom-based virial contribution of all covalent (including bond con-straints) and nonbonded (excluding reciprocal-space lattice-suminteractions) terms is calculated using eq. (18) or one of itsgeneralizations.

The GROMOS05 implementation (both PROMD and MD��)includes the possibility of using a group-based pressure definition(corresponding to any arbitrary repartition of atoms into virialgroups), instead of the atom-based one. In this case, the intragroupcontribution to the kinetic energy as well as the contribution ofintragroup forces to the virial are removed from the pressuredefinition (which affects the fluctuations of this quantity, but not itsaverage value). As shown elsewhere13,14 (the equations reportedtherein should be altered by halving the virial and replacing rij by�rij to match the present conventions), the group-based virialtensor can be calculated from the corresponding atom-based tensorby adding a simple correction term, which depends on the overallatomic forces and on the virial groups definitions. More precisely,the group-based virial tensor is given by


� �* �1

2 �I�

FI� � dI�, (19)

where I� denotes atom � in virial group I, FI� the overall force onatom I�, and dI� the coordinate vector of atom I� relative to thecenter of mass of the gathered virial group I containing this atom.The “gathered” representation of the virial group is generated byfollowing the atoms as they drift throughout the periodic system.The group-based pressure tensor is then calculated as

� 2

�� , (20)

where � is the group-based internal kinetic energy tensor, definedas

� 1

2 �I�1

Ns ��1

Na�I�

mi��1��1

Na�I�

miri� � ��1

Na�I�

miri�, (21)

where Ns is the number of virial groups and Na(I) the number ofatoms in virial group I.

Although the atom-based pressure tensor �* is typically sym-metric, this is generally not the case for the group-based pressuretensor � (although the antisymmetric contribution to this tensorshould vanish upon time averaging). When applying a barostatalgorithm, the antisymmetric component of � should induce anoverall rotation of the computational box (which would alter thebox angular momentum), while the symmetric component resultsin a deformation of the box (which conserves the box angularmomentum). In practice, the overall rotation of the box is rather anuisance, and is avoided by symmetrizing the tensor (� 3(1/ 2)[� � t�]) prior to application of the barostat algorithm,18

where the “t” presuperscript indicates the transpose of the matrix.

Thermostat Algorithms

MD simulation relies on integrating the classical (Newtonian)equations of motion for a molecular system, and thus, samples amicrocanonical (constant-energy) ensemble by default. However,for compatibility with the experiment, it is often desirable tosample configurations from a canonical (constant-temperature)ensemble instead. A modification of the basic MD scheme with thepurpose of maintaining the temperature constant (on average) iscalled a thermostat algorithm.7 Note that in contrast, SD automat-ically generates a canonical ensemble, at a temperature determinedby the balance between the magnitudes of the random and fric-tional forces.

In the GROMOS05 implementation (both PROMD andMD��), four different thermostat algorithms are available: (1)temperature constraining (Woodcock thermostat19); (2) tempera-ture relaxation by weak coupling (Berendsen thermostat20); (3)temperature relaxation by an extended-system method (Nose–Hoover thermostat21,22); (4) temperature relaxation by the Nose–Hoover chain thermostat.23 In all cases, the instantaneous temper-ature � is calculated as described earlier, and relaxed towards a

temperature To associated with the heat bath to which the systemis coupled. The three latter algorithms also involve the specifica-tion of the characteristic time � for this relaxation. All the abovethermostat algorithms rely on a scaling of the atomic velocitiesafter each integration time step. This scaling should only operateon the internal velocities, excluding any component along theexternal degrees of freedom (see earlier). Due to the constraintsimposed in the GROMOS05 implementation on the system totallinear and angular momenta, the internal velocities only differfrom the real ones when MD is applied under vacuum boundaryconditions. In this case, the velocity scaling is applied on theinternal velocities ri and the real velocities ri

o can be recoveredthrough the inverse of eq. (11), namely

rio ri � I�CM

�1 �ro�Lsyso �ri

o rCMo �. (22)

When simulating molecular systems involving distinct sets ofdegrees of freedom with either (1) very different characteristicfrequencies or (2) very different heating (or cooling) rates causedby algorithmic noise (e.g., electrostatic cutoff, application of atom-distance constraints), the joint coupling of all degrees of freedomto a single thermostat may lead to different effective temperaturesfor the different sets of degrees of freedom (due to a too slowexchange of kinetic energy). A typical example is the so-called“hot solvent–cold solute problem” in simulations of macromole-cules. Because the solvent is often more significantly affected byalgorithmic noise (heating due to the use of an electrostatic cutoff),the coupling of the whole system to a single thermostat may causethe average solute temperature to be significantly lower than theaverage solvent temperature. In the GROMOS05 implementation(both PROMD and MD��), this problem may be alleviated byseparately coupling different subsets of degrees of freedom (e.g.,solute, counterions, cosolvent, and solvent) to different indepen-dent thermostats.

The prototype of most isothermal equations of motion is

ri�t� mi�1Fi�t� ��t�ri�t�. (23)

The function �(t) controls the heat exchange between the systemand the heat bath. A negative value indicates that heat flows to thesystem, while a positive value indicates a heat flow in the oppositedirection. Practical implementations of eq. (23) rely on the step-wise integration of Newton’s second law [eq. (23) with �(t) � 0],altered by the scaling of the atomic velocities after each iterationstep. In the context of the leapfrog integrator24 used in GRO-MOS05, this can be written as

ri�t ��t

2 � ��t; �t�r�i�t ��t

2 � ��t; �t��ri�t

�t

2 � � mi�1Fi�t��t�, (24)

where �(t; �t) is a time- and time step-dependent velocity scalingfactor. Imposing the constraint �(t; 0) � 1, one recovers eq. (23)in the limit of an infinitesimal time step �t, with


��t� ��(t; �t)

�(�t)�

�t�0

. (25)

The Woodcock thermostat19 (also known as temperature con-straining thermostat; see also the Hoover–Evans thermostat25,26)aims at fixing the instantaneous temperature � exactly at thereference heat-bath value To, without allowing for any fluctua-tions. In this case, the quantity �(t; �t) in eq. (24) is found byimposing �[t � (�t/ 2)] � ( g/Ndf)To, leading to

��t; �t� �g

Ndf

To

��t ��t

2 �1/ 2

, (26)

where ��[t � (�t/ 2)] is the temperature evaluated from thevelocities r�i[t � (�t/ 2)] in eq. (24). The corresponding quantity�(t) in eq. (23) is given by

��t� � gkBTo��1 �

i�1

N

ri�t� � Fi�t�. (27)

Although g � Ndf seems to be the obvious choice, it turns out thatg � Ndf � 1 is the appropriate one for the algorithm to generatea canonical distribution of configurations (although obviously notof momenta) at temperature To.7,21,25 The reason is that constrain-ing the temperature effectively removes one degree of freedomfrom the system. Note, however, that the absence of kinetic energyfluctuations may lead to inaccurate dynamics, especially in thecontext of the microscopic systems typically considered in simu-lations.

The Berendsen thermostat20 (also known as weak-couplingthermostat) aims at relaxing the instantaneous temperature � to thereference heat-bath value To based on a first-order (weak-cou-pling) scheme with a characteristic time �B, that is, as

��t� �B�1�To ��t�. (28)

In this case, the quantity �(t; �t) in eq. (24) is found by imposing�[t � (�t/ 2)] � �[t � (�t/ 2)] � �B

�1�t( g/Ndf){To � �[t �(�t/ 2)]}, where, in principle, g � Ndf, leading to

��t; �t� �� t �t

2 �� t �

�t

2 �� B

�1�t

g

NdfTo ��t

�t

2 ��t �

�t

2 � �1/ 2

� 1 � �B�1�t�

g

NdfTo

��t ��t

2 � 1�

1/ 2

. (29)

In GROMOS05, the algorithm is implemented following the sec-ond (approximate) expression. For either of the two expressions,the corresponding quantity �(t) in eq. (23) is given by

��t� 1

2�B

�1� g

Ndf

To

�(t) 1�. (30)

In practice, �B is used as an empirical parameter to adjust thestrength of the coupling to the heat bath. In the limit �B � �t, theBerendsen algorithm is equivalent to the Woodcock algorithm(and thus generates a canonical distribution of configurations, butnot of momenta). In the limit �B 3 , the thermostat becomesinactive and the Newton equation of motion is recovered (whichsamples a microcanonical ensemble). However, except in theformer limit (and only for the configurational part), the ensemblegenerated by the Berendsen equations of motion is not a canonicalensemble.27

The Nose–Hoover thermostat21,22 aims at relaxing the instan-taneous temperature � to the reference heat-bath value To basedon an extended-system approach with a characteristic time �NH. Inthe original Nose algorithm,28 the real system is extended byaddition of an artificial (Ndf � 1)th (positive) dynamical variable s(associated with a “mass” Q � 0 as well as a velocity s), thatplays the role of a time-scaling parameter. Through an appropriatechoice for the extended-system Lagrangian, a microcanonical MDtrajectory in the extended system can be mapped onto a canonicaltrajectory in the real system. However, the Nose thermostat leadsto sampling of the real-system trajectory at uneven time intervals,which is quite impractical. This inconvenience is alleviated byrewriting the equations of motion in terms of the real-systemvariables, as was later shown simultaneously by Nose21 andHoover.22 In the Nose–Hoover algorithm, the quantity �(t) in eq.(23) is not uniquely determined by the instantaneous microstate ofthe system, but is a dynamical variable where the derivative isdetermined by this instantaneous microstate through

� ��NH�2

�

To� g

Ndf

To

� 1�, (31)

where the effective coupling time �NH is related to the (lessintuitive) effective mass Q in the Nose thermostat through

�NH �NdfkBTo��1/ 2Q1/ 2. (32)

When � is negative, heat flows from the heat bath into the systemdue to eq. (23). When the system temperature increases above To,the time derivative of � becomes positive in eq. (31) and the heatflow is progressively reduced (feedback mechanism). Conversely,when � is positive, heat is removed from the system until thesystem temperature decreases below To and the heat transfer isslowed down. In practice, eq. (31) is discretized (based on thesimulation time step �t) and integrated simultaneously with theequations of motion for the atomic coordinates and velocitiesbased on the leapfrog scheme.

It can be proven7,22 that the Nose–Hoover equations of motionsample a canonical ensemble (in both coordinates and momenta)provided that g � Ndf and that �NH is finite, this irrespective of theactual value of �NH and of the initial conditions for the atomicvelocities and for the � variable.

In practice �NH is used as an empirical parameter to adjust thestrength of the coupling to the heat bath. Too large values of �NH


(loose coupling) may cause a poor temperature control (the limit-ing case of the Nose–Hoover thermostat with �NH3 and �(0) �0 is MD, which generates a microcanonical ensemble). On theother hand, too small values (tight coupling) may cause high-frequency temperature oscillations leading to the same effect.

The Nose–Hoover chain thermostat23 aims at relaxing theinstantaneous temperature � to the reference heat bath value To

based on a chain of successive thermostat variables. In this case thesingle thermostat variable � of the Nose–Hoover scheme is re-placed by a chain of variables applying a thermostat to each otherin sequence. This algorithm has been introduced to alleviate thetwo main drawbacks of the Nose–Hoover algorithm: (1) the pres-ence of temperature oscillations, and (2) the nonergodicity of thesampling for small or stiff systems, or systems at low tempera-tures.22,29–34 The GROMOS05 implementation follows the for-malism described in the original article.23

Barostat Algorithms

For compatibility with the experiment, it is often desirable tosample configurations from the isothermal–isobaric ensemble(constant temperature and pressure). Thermostat algorithms havebeen described above. A modification of the basic MD schemewith the purpose of maintaining the pressure constant (on average)is called a barostat algorithm.

The use of a barostat is only applicable to simulations underperiodic boundary conditions. In the GROMOS05 implementation(both PROMD and MD��), the various options for the variationsfor the box parameters (and the associated scaling of atomiccoordinates) involved in the use of a barostat are: (1) no variationsof the box parameters; (2) isotropic scaling, that is, identicalrelative variations of the box-edge lengths only; (3) partiallyanisotropic scaling, that is, independent relative variations of thebox-edge lengths only; (4) fully anisotropic scaling, that is, inde-pendent variations of all box parameters (box-edge lengths, boxangles, and Euler angles). For a truncated-octahedral box, only thefirst two options are allowed. For a rectangular box, only the firstthree options are allowed. For a triclinic box, all options areallowed. In the latter case, variations in the box shape are accom-panied by variations in the box Euler angles, so as to guarantee thatthe barostat does not introduce a rigid-body rotational componentto the box orientation. Note, however, that the location of the boxcenter of mass is affected by any type of coordinate scaling.

Two different barostat algorithms are available (1) pressurerelaxation by weak-coupling (Berendsen barostat20); (2) pressurerelaxation by extended-system method (Andersen–Parrinello–Rah-man barostat21,28,35–40).

Pairlist Construction

The evaluation of the nonbonded interactions in GROMOS relieson the application of the twin-range method.41–44 The GRO-MOS05 implementation (both PROMD and MD��) of this ap-proach includes an increased amount of flexibility, and relies onthe definition of: (1) a short-range pairlist distance Rp; (2) acorresponding cutoff distance Rp � Rp (optional); (3) a lowerbound for the intermediate-range pairlist distance Rs; (4) a corre-sponding cutoff distance Rs � Rs (optional); (5) an upper bound

for the intermediate-range pairlist distance Rl; (6) a correspondingcutoff distance Rl � Rl (optional); (7) a short-range pairlist updatefrequency Ns; (8) an intermediate-range pairlist update frequencyNl. Short-range interactions are computed every time step basedon a short-range pairlist containing pairs in the distance range [0;Rp], or a filtered subset of this list corresponding to pairs currently(i.e., at the given timestep) in the distance range [0; Rp]. Theshort-range pairlist is reevaluated every Ns time steps. It can begenerated either on the basis of distances between charge groups(groups of covalently linked atoms defined in the system topology)or of distances between individual atoms. In the former case, thefiltering (based on the distance Rp) may be based either on dis-tances between charge groups or on distances between atoms. Inthe latter case, only atom-based filtering is possible. Intermediate-range interactions are computed every Nl time steps based on allpairs in the distance range [Rs; Rl], or a filtered subset of thesepairs in the distance range [Rs; Rl] at the time of the evaluation ofthese interactions. Only an atom-based filtering is possible here,and it is only meaningful when the initial set of pairs is generatedon the basis of distances between charge groups. The energy,forces, and virial contributions associated with intermediate-rangeinteractions are assumed constant between two updates (i.e., dur-ing Nl steps).

The evaluated interaction includes Lennard–Jones and electro-static components. The latter component may include a reaction-field contribution or the real-space contribution to a lattice-summethod. Note that the real-space contribution to a lattice-summethod may only be computed within the short-range contributionto the interaction.

The pairlist construction may be performed in four differentways: (1) using the standard double-loop algorithm included in theGROMOS96 program3 (merely extended to include the possibilityof an atom-based cutoff and of filtering); (2) using an optimizedversion that improves processor cache usage; (3) using a grid-based pairlist algorithm introduced recently45 (PROMD only); (4)using a slight variation of the above grid-based algorithm,45 whichpermits easier parallelization and avoids periodicity correctionsduring the interaction evaluation (MD�� only).

Pairlist construction with optimized processor cache usage. Animproved version of the standard double-loop algorithm is imple-mented in PROMD, which tries to optimize processor cache usage.Figure 2 illustrates the basic idea for a coordinate set of N � 10atoms (or charge groups). In a double loop running over atomindices, the standard algorithm of GROMOS96 scans the atom pairmatrix row by row to decide whether a given pair of atoms needsto be included in the pairlist (unshaded box). For very large N, thefast but small processor cache obviously needs to be reloadedmany times. If we assume that this cache can hold the coordinatesof Ncache atoms at a time, then a total of Ncache � 1 atom pairs canbe treated per cache reload. Scanning the atom-pair matrix withquadratic (light gray shade) or rhombic (dark gray shade) windowsis obviously more efficient, because O(Ncache

2 ) atom pairs can betreated per cache reload. The triangular matrix of atom pairs (Fig.2, top) can be reordered into a rectangular one by simply aligningevery row to the left, chopping off the right half and appendingeach of its columns to the rows of the lower portion of the matrix


as shown in Figure 2 (bottom). In practice, the same rectangularmatrix is more conveniently generated from two coordinate vec-tors, the second one being twice as long as the first one andcontaining, consecutively, two identical copies of the first one (i.e.,1, 2, 3, . . . , N � 1, N, 1, 2, 3, . . . , N � 1, N). To generatethe entire rectangular matrix, one simply needs to shift the twovectors with respect to each other, by 1 for the first column, by 2for the second, and so on. The formerly rhombic window becomesquadratic in the reordered matrix, and all rows are of equal length,which makes the algorithm simpler. Some consideration is neces-sary to handle the last column adequately, because for even N itcontains each atom pair in duplicate. The new algorithm requiressomewhat more memory than the traditional implementation as itmaintains three copies of the same coordinate set. In principle,however, it can handle a total of (Ncache � 1)2/nine atom pairs(rather than Ncache � 1) in a given amount of fast processor cache.In practice, along with additional code optimization but also ad-ditional postprocessing necessary to generate properly orderedpairlists, we observe speedup factors of about 2–3 for the pairlistconstruction in applications to medium-sized and large systems onpersonal computers (see Table 2 for overall efficiency; timings aregiven for complete simulations including pairlist construction andforce calculation). A shared-memory OpenMP (www.openmp.org)-style parallel version of the algorithm has also been imple-mented, which distributes the workload, window by window, overseveral processors.

Grid-based pairlist construction. PROMD includes a recently in-troduced45 grid-based pairlist algorithm that permits the fast con-struction of cutoff-based nonbonded pairlists in molecular simu-

lations under periodic boundary conditions based on an arbitrarybox shape (rectangular, truncated-octahedral, or triclinic). The keyfeatures of this algorithm are: (1) the use of a one-dimensionalmask array (to determine which grid cells contain interactingatoms) that incorporates the effect of periodicity, and (2) thegrouping of adjacent interacting cells of the mask array intostripes, which permits the handling of empty cells with a very lowcomputational overhead. Testing of the algorithm on water sys-tems of different sizes (containing about 2000 to 11,000 mole-cules) has shown that the method: (1) is about an order of mag-nitude more efficient compared to a standard (double-loop)algorithm, (2) achieves quasi-linear scaling in the number ofatoms, (3) is weakly sensitive in terms of efficiency to the chosennumber of grid cells.

MD�� includes a slightly modified version of this grid-basedpairlist algorithm following ideas similar to those of a publishedpairlist algorithm.46,47 In an effort to reduce the number of nearestimage determinations during the pairlist generation (or filtering)and the nonbonded force calculation, the system gets extended onall sides before the pairlist construction. The additional atom orcharge-group positions are obtained by simple shifts of the originalpositions by the box vectors. To allow for more efficient (distrib-uted-memory) parallelization and to save money, the central com-putational box is divided into N layers. Each of the P parallelprocesses only has to extend over N/P layers. After every exten-sion, the atom pairs consisting of one atom within the layer and asecond atom from one of the above (not extended) layers are addedto the respective pairlist (using a one-dimensional mask array).Filtering or energy and force evaluation can then be carried outright away (without nearest image determinations owing to thepreshifted atomic positions), or at a later stage with the informationon the shift vectors encoded into the pairlist, thus enabling a fastreconstruction of the shifted positions.

Reaction-Field Electrostatics

When cutoff truncation is applied to the Coulombic interactionswithin a molecular system, the mean effect of the omitted electro-static interactions beyond the (long-range) cutoff distance Rl maybe approximately reintroduced through a so-called reaction-fieldcorrection term.48–51 This approximation relies on assuming thatthe medium beyond the cutoff sphere of each particle (i.e., beyonda specified distance RRF, typically set equal to Rl) is a linearizedPoisson–Boltzmann continuum characterized by a relative dielec-tric permittivity � and an inverse Debye screening length �. In thepresent context, these two parameters may be combined into aneffective permittivity51

�RF �1 �(�RRF)

2

2(�RRF � 1)��. (33)

In the GROMOS05 implementation (both PROMD andMD��), the corresponding overall electrostatic energy (Coulombplus reaction-field term) is then written52

�elCB�RF

1

4��o �i

�j�i, j�excl�i�,r� ij�Rl

qiqj�r�ij�1 �

2(�RF 1)

2�RF � 1

r�ij2

2RRF3

Table 2. Efficiency Comparison of GROMOS96 [FORTRAN, StandardPairlist Algorithm (STD), and Pairlist Algorithm with Optimized CacheUsage (OPT)] and MD�� [C��; Standard Pairlist Algorithm (STD),and Grid-Based Pairlist Algorithm (GRID)].

GROMOS96PROMD MD��

STD OPT STD GRID

Alkane Single 166 102 214 49 (45)Dual — 61 121 40 (30)

Membrane Single 67 57 95 55 ( )Dual — 34 60 43 (29)

Protein Single 175 148 349 140 ( )Dual — 82 187 90 (64)

As test systems liquid alkane (23,328 solute atoms, 9.4 � 9.4 � 9.4 nmcubic box), a membrane [6656 solute atoms, 7383 solvent (SPC water)atoms, 6.2 � 6.2 � 6.9 nm rectangular box], and a protein [2445 soluteatoms, 47,472 solvent (SPC water) atoms, 7.9 � 7.9 � 8.3 nm rectangularbox] were used. All calculations were done on a dual-processor AMDAthlon MP 248 PC (2000-MHz processor frequency, 512-kb cache, 2-GBRAM). The efficiency was measured running on a single processor andrunning in parallel on both processors, 250 simulation steps for the alkaneand membrane system, 100 steps for the protein. All numbers are inseconds. For MD��, the time spent doing interaction calculations is givenin parentheses.


3�RF

2�RF � 1

1

RRF� � �i

�j�i, j�excl�i�

qiqj�2(�RF 1)

2�RF � 1

r�ij2

2RRF3

3�RF

2�RF � 1

1

RRF� 1

2

3�RF

2�RF � 1

1

RRF ��i

qi2 �RF

�1��i

qi�2��,

(34)

where �o is the permittivity of vacuum, r�ij is the minimum-imagevector corresponding to rij � ri � rj, and excl(i) denotes theexclusion list of atom i (including its first and second covalentneighbors; the distance between any two excluded atoms is as-sumed to be smaller than Rl). Note that current simulation pro-grams (e.g., GROMOS963 and GROMACS53) typically restrict theimplementation of the reaction-field method to the first term in eq.(34). The second term is explicitly included here because excludedneighbors should only be exempted from the direct (Coulombic)interaction and not from the solvent-mediated (reaction-field) in-teraction.52 The form of the third term has been chosen for con-sistency in the context of small molecules (compared to the cutoffradius and box size). For such a small molecule (or ion) gatheredby periodicity around its center, r� ij can be replaced by rij and thecutoff truncation involved in the first summation of eq. (34) can beomitted. In this case, it can be shown52 that the reaction-fieldcontribution to �el

CB�RF for a neutral molecule matches the correctOnsager expression for the solvation of a dipolar molecule in aspherical cavity54 (for � � 0). For a monoatomic ion, the last termcan also be shown52 to represent a (first-order) correction to theerror in solvation free energy caused by the use of effective(non-Coulombic) interactions. Intuitively, this last term may beinterpreted as the reversible work required to individually chargethe atoms when they are at infinite separation. This contributiononly affects the energy of the system, but does not induce atomicforces. However, it may be essential to include it in free-energycalculations involving alterations of the atomic partial charges andcomparisons between different media (�).

Lattice-Sum Electrostatics

Lattice-sum methods for evaluating electrostatic interactions insimulations under periodic boundary conditions (only available inPROMD) rely on two key principles: (1) the treatment of electro-static interactions as exactly periodic within the periodic sys-tem.55–58 (2) the splitting of these interactions into a short-rangecomponent, evaluated by direct summation over atom pairs, and along-range component, evaluated by Fourier series.56,59,60

From a physical point of view, the straight application oflattice-sum methods (with so-called tinfoil boundary conditions,that is, omitting any extrinsic potential contribution) implicitlyrelies on four modifications of electrostatic interactions comparedto the naive picture of assembling an infinite number of copies ofthe reference computational box in vacuum:60 (1) inclusion of ahomogeneous background charge density within the infinite peri-odic system (of magnitude related to the net charge of the com-putational box), that enforces a vanishing net charge in the sys-tem;55 (2) inclusion of a surface charge (with a distribution relatedto the box dipole moment relative to its center) at the surface of the

infinite periodic system, that enforces a vanishing average electricfield within the system; (3) inclusion of a homogeneous surfacedipole layer (of magnitude related to the trace of the box quadru-pole moment relative to its center) at the surface of the infiniteperiodic system, that enforces a vanishing average potential withinthe system; (4) suppression of the orientational preferences result-ing from a fluid–vacuum interface at the surface of the system,which is effectively equivalent to the inclusion of a compensatinghomogeneous dipole layer at the surface of the infinite periodicsystem. These physical modifications have subtle consequences onthe simulated properties of molecular systems, which can be al-tered (on the basis of physical arguments) by the introduction of aso-called extrinsic potential contribution (consisting of a uniformelectric field and a constant potential offset). However, the appro-priate form for this extrinsic term is still matter of considerabledebate.56,58,60–82 The PROMD implementation for this term isdiscussed elsewhere.60

The splitting of the electrostatic interactions into a short-rangeand a long-range component relies on the use of a charging-shaping function a�3�(a�1r) of width a. The charge-shapingfunctions implemented in PROMD are the Gaussian and optimizedtruncated polynomials of orders 0 to 10,59 as listed in Table 3. Thespecific choice of a function must be made before compiling theprogram (macro definition).

The two lattice-sum methods available differ in the way theyevaluate the reciprocal-space interactions.56 The Ewald method83

is based on a direct summation over reciprocal-space lattice vec-tors, while the particle–particle–particle–mesh (P3M) method84

makes use of grid discretization and fast Fourier transform (FFT)algorithms. In the PROMD implementation, both lattice-summethods can also be applied to emulate truncated electrostaticinteractions with a reaction-field correction, according to the latticesum emulated reaction field (LSERF) scheme.60

Reciprocal lattice. In the triclinic case, the reciprocal-lattice vec-tors a, b, and c associated with the box edge vectors a, b, and c aredefined by

a V�1b c, b V�1c a and c V�1a b, (35)

where V is the box volume. The matrix containing in its columnsthe components of a, b, and c in the reference Cartesian coordinatesystem (ex, ey, ez) is easily shown to be

� ax bx cx

ay by cy

az bz cz

� tL� �1 R� tS� �1. (36)

A reciprocal-space vector k is defined by

k 2��laa � lbb � lcc�, (37)

where l � (la, lb, lc) is a vector with integer (positive or negative)components. Based on a general triclinic box in an arbitraryorientation, a reciprocal-space vector may be specified through theinteger vector l, through oblique fractional coordinates � � (�u,�v, �w) with reference to the reciprocal-lattice vectors, or through


coordinates k � (kx, ky, kz) within the reference Cartesiancoordinate system. The different coordinates are related through

� 2�l and k tL� �1� 2�tL� �1l. (38)

Scalar products between real- and reciprocal-space vectors can beformulated equivalently in the different coordinate representations,that is,

k � r � � �. (39)

Definitions. The following discussion considers a periodic systemof Nq charges qi at positions ri within a general triclinic compu-tational box with arbitrary orientation. It is convenient to define thebox overall charge

S �i�1

Nq

qi, (40)

and the box overall square charge

S2 �i�1

Nq

qi2. (41)

Lattice-sum methods rely on the use of the charge-shapingfunction a�3�(a�1r) of width a (Table 3) to split the electrostaticpotential into a real-space contribution and a reciprocal-spacecontribution, plus a constant. The charge-shaping function is nor-malized to satisfy the condition

4�a�3 �0

drr2��a�1r� 1. (42)

The following definitions are related to the charge-shaping func-tion. The Fourier coefficients of (a lattice sum of) the charge-shaping function are given by

��ak� 4�k�1a�3 �0

drr sin�kr��a�1r� for k � 0

1 for k 0

. (43)

The switch function �(a�1r) associated with the charge-shapingfunction is defined by

��a�1r� 4�a�3 �r

d�� r��a�1��. (44)

Finally, the constants A1, A2, and A3 are defined as

A1 �4�V�1 �0

drr��a�1r�, (45)

A2 4�V�1 �l��3,l�0

k�2��ak�, (46)

A3 limr30

� �n��3

�r � L� n��1�(a�1�r � L� n�) r�1�. (47)

In the (common) case where

��a�1r� 0 for r � min�Lx, Ly, Lz� (48)

is (exactly or approximately) valid, eq. (47) becomes

Table 3. Charge-Shaping Functions Available in PROMD.

N� m ��(�)

�1 ��1/2e��2

0 0 (3/4)H(1 � �)1 1 3(1 � �)H(1 � �)2 2 (15/2)(1 � �)2H(1 � �)3 2 (15/4)(1 � �)2(1 � 2�)H(1 � �)4 3 (105/16)(1 � �)3(3� � 1)H(1 � �)5 4 (21/2)(1 � �)4(4� � 1)H(1 � �)6 4 (63/8)(1 � �)4(5�2 � 4� � 1)H(1 � �)7 5 (45/4)(1 � �)5(8�2 � 5� � 1)H(1 � �)8 6 (165/32)(1 � �)6(35�2 � 18� � 3)H(1 � �)9 6 (165/64)(1 � �)6(64�3 � 69�2 � 30� � 5)H(1 � �)

10 7 (2145/128)(1 � �)7(21�3 � 19�2 � 7� � 1)H(1 � �)

N�: reference code of the function (N� � 0 . . . 10: Optimal TP-Function59 of order N�; N� � �1:Gaussian); m: convergence rate of �(�) toward zero (convergence is as ��(m�2) when � 3 ; seeeq. (43), where � � ak); ��(�): charge-shaping function amplified by � (the actual shaping functionis a�3�(a�1r)); H(�): Heaviside function (H(�) � 1 when � � 0, zero otherwise). Note that theGaussian function is infinite-ranged, while all other functions vanish for � � 1.


A3 limr30

r�1��a�1r� 1. (49)

In this case, A1 and A3 have analytical expressions for a givencharge-shaping function. The above derived quantities56,59 arelisted in Tables 4, 5, and 6 for the charge-shaping functions ofTable 3.

Electrostatic interaction energy. The electrostatic reversible-charging (free) energy �el

LS of the periodic system of charges undertinfoil boundary conditions can be computed as56

�elLS �� A � �

slf,(50)

with

�� 2�oV��1 �i�1

Nq �j�1

Nq

qiqj �l��3,l�0

k�2��ak�cos�k � rij�, (51)

�� 4��o��1 �

i�1

Nq �j�i

Nq

qiqj �n��3

�rij � L� n��1��a�1�rij � L� n��,

(52)

�A �8��o��1�A1S

2 � A1 � A2�S2, (53)

Table 4. Switch Functions Corresponding to the Charge-Shaping Functions Available in PROMD(see Table 3).

N� �(�)

�1 erfc(�)0 (1/2)(1 � �)2(� � 2)H(1 � �)1 (1 � �)3(� � 1)H(1 � �)2 (1/2)(1 � �)4(3� � 2)H(1 � �)3 (1/4)(1 � �)4(4�2 � 7� � 4)H(1 � �)4 (1/8)(1 � �)5(15�2 � 19� � 8)H(1 � �)5 (1 � �)6(3�2 � 3� � 1)H(1 � �)6 (1/16)(1 � �)6(35�3 � 66�2 � 51� � 16)H(1 � �)7 (1/8)(1 � �)7(32�3 � 49�2 � 31� � 8)H(1 � �)8 (1/16)(1 � �)8(105�3 � 136�2 � 73� � 16)H(1 � �)9 (1/32)(1 � �)8(160�4 � 335�3 � 312�2 � 151� � 32)H(1 � �)

10 (1/128)(1 � �)9(1155�4 � 2075�3 � 1665�2 � 697� � 128)H(1 � �)

N�: reference code of the function; �(�): switch function (the real-space interaction function isr�1�(a�1r)); erfc: complementary error function.

Table 5. Fourier Coefficients Corresponding to the Charge-ShapingFunctions Available in PROMD (see Table 3).

N� �N��3�(�)

�1 �2e��2/4

0 3[��C � S]1 12[2 � 2C � �S]2 60[2� � �C � 3S]3 90[8 � (�2 � 8)C � 5�S]4 630[8� � 7�C � (�2 � 15)S]5 5040[4(�2 � 6) � (�2 � 24)C � 9�S]6 7560[48� � (�2 � 57)�C � 3(4�2 � 35)s]7 75600[24(�2 � 8) � 3(5�2 � 64)C � (�2 � 87)�S]8 831600[8�(�2 � 24) � (�2 � 123)�C � (18�2 �

315)S]9 1247400[192(�2 � 10) � (�4 � 207�2 � 1920)C �

(22�2 � 975)�S]10 16216200[64�(�2 � 30) � (26�2 � 1545)�C � (�4 �

285�2 � 3465)S]

N�: reference code of the function; �N��3�(�): Fourier coefficient ampli-fied by �N��3 (the actual Fourier coefficient is �(ak)); C: short notationfor cos(�); S: short notation for sin(�).

Table 6. A-Constants Corresponding to the Charge-Shaping FunctionsAvailable in PROMD (See Table 3).

N� �V��1a�2A1 �aA3

�1 1 2��1/2

0 2/5 3/21 4/15 22 4/21 5/23 3/14 9/44 1/6 21/85 2/15 36 8/55 45/167 4/33 25/88 4/39 55/169 10/91 105/32

10 2/21 455/128

N�: reference code of the function; �V��1a�2A1: constant A1 amplifiedby �V��1a�2; �aA3: constant A3 amplified by �a. The results for A3

are derived assuming the validity of eq. (48).


and

�slf �8��o��1�A1 � A2 � A3�S

2. (54)

The interpretation of the different contributions to �el in eq.(50) is given below, with reference to the following terminologyfor charge densities (which can be added to or subtracted from oneanother): point charge ( p), �-shaped charge (�), and homogeneousbackground charge (b). The term �� represents the electrostaticenergy (including self-interaction) of a set of ( p � b)-charges ofmagnitude {qi} at positions {ri} in the potential generated by thecorresponding periodic system of (� � b)-charges. Because thispotential is a nonsingular and generally smooth function of posi-tion, �� is conveniently evaluated in reciprocal space, using theEwald method83 or the P3M method.84 The term �� represents theelectrostatic energy (excluding self interaction) of a set of ( p �b)-charges of magnitude {qi} at positions {ri} in the potentialgenerated by the corresponding periodic system of ( p � �)-charges. With an appropriate choice of charge-shaping function,the function �(a�1r) can be made a quickly decreasing function ofdistance, in which case �� is conveniently evaluated by direct(real-space) summation over the charge pairs. The terms �A and�slf are configuration-independent. The term �A eliminates theself-energies present in �� and contains a small correction due tothe constraint of zero average potential within the periodic system.The term �slf accounts for the self-energy of set of ( p � b)-charges of magnitude {qi} in the potential generated by the corre-sponding periodic system of (p � b)-charges (Wigner term83–87).

Constant and self-energy terms. The constant term �A and theself-energy term �slf are given by eqs. (53) and (54), respectively,where the A-constants are defined by eqs. (45), (46), and (47) or(49). Note that these terms give rise to no force contribution (buta virial contribution).

In the general case, the constant A2 must be computed numer-ically by direct summation. In the PROMD implementation, this isdone by evaluating

A2�lmax� 4�V�1 �l��3,l�0,lx,ly,lz�lmax

k�2��ak� (55)

for increasing values of lmax, until a user-specified relative toler-ance is reached. In the specific case of a cubic unit cell of edge L,this evaluation is replaced by49,87 A2 �EWL�1 � A1 � A3 with�EW �2.83792748. The calculation of A2 with a reasonablyhigh precision is somewhat expensive and may be either (1)omitted; (2) performed once at the beginning of the simulation; (3)performed whenever an energy output is required; (4) performedevery step. With the first choice, A2 is set to zero and the A2

contributions are omitted in the evaluation of both �A and �slf

[eqs. (53) and (54)]. As a consequence, the splitting betweenpairwise and self-contributions becomes arbitrary, but the sum ofthe two quantities (and thus the overall electrostatic energy) re-mains correct (within the approximation A2 A2, see below). Thesecond choice is intended for constant-volume simulations. Thetwo last choices are for constant-pressure simulations, the latter

one being the more accurate (but also computationally more ex-pensive).

The quantity A2 calculated through eq. (55) represents the exact(in the limit of large lmax) value of A2 used in the calculation of�slf . However, because the reciprocal-space interaction energy isevaluated with a finite precision (i.e., through the Ewald or P3Mmethod), this value of A2 will only approximately remove thereciprocal-space self-energy when included in �A . Although formost practical purposes, this approximation is sufficient, it ispossible to compute a more accurate method-dependent value A2

to be used in the evaluation of �A, that is,

�A �8��o��1�A1S

2 � A1 � A2�S2, (56)

The calculation of this quantity is feasible for either the Ewald orP3M method, as detailed elsewhere.13

Real-space contribution. In the most general form, the real-spacecontribution to the electrostatic interactions is given by eq. (52),together with the corresponding forces and virial contribution. Inpractice, it is assumed that the charge-shaping function satisfies thecondition

��a�1r� 0 for r � Rp with Rp � �1/2�min�Lx, Ly, Lz�,

(57)

where Rp is the short-range cutoff distance applied to real-spaceinteractions, which implies that the switch function �(a�1r) alsovanishes beyond Rp. In this case, and taking into account thatCoulombic interaction between minimum-image pairs of excludedcovalent neighbors should be removed, one may rewrite eq. (52) as

�� 4��o��1 �

i�1

Nq �j�i, j�excl�i�,r� ij�Rp

Nq

qiqjr�ij�1��a�1r�ij�

� �4��o��1 �

i�1

Nq �j�i, j�excl�i�

Nq

qiqjr�ij�1��a�1r�ij� 1, (58)

where the distance between any two excluded atoms is assumed tobe smaller than Rp. When using a truncated-polynomial charge-shaping function,59 the evaluation of �� is exact provided that allatom pairs within a distance smaller than a are included into theshort-range pairlist at any time step. When using a Gaussianfunction, it is always approximate (in practice a Rp/3 is areasonable choice).

Reciprocal-space contribution via Ewald. In the Ewald method,83

the reciprocal-space contribution �� defined by eq. (51), as well asthe corresponding forces and virial, are evaluated by direct sum-mation over reciprocal-lattice vectors.

For improved computational speed the triple-sum (over l, i, andj) in eq. (51) is rewritten as a double-sum (over l and i) andtruncated at a given reciprocal-space cutoff lc, as


�� 2�oV��1 �l�lc

k�2��ak��C2�k� � S2�k�, (59)

with the definitions

C�k� �i�1

Nq

qicos k � ri and S�k� �i�1

Nq

qisin k � ri. (60)

A further increase in computational efficiency can be obtained bynoting that the terms in the l-sum involved in eq. (59) are invariantupon changing k to �k. Thus, the summation can be restricted toa half-space and the resulting energies, forces, and virial contri-bution multiplied by 2. In the case of a rectangular box, furthersymmetry considerations allow to restrict the summation to asingle octant.

Reciprocal-space contribution via P3M. In the P3M method,84 thereciprocal-space contribution �� defined by eq. (51), as well as thecorresponding forces and virial, are evaluated by discretization ofthe triclinic computational box by means of a grid (mesh) and useof a FFT algorithm. The number of grid subdivisions along each ofthe box axes must be even.

The algorithm consists of six steps:56 (1) assignment of thecharge density associated with the atomic partial charges to thegrid points by means of an assignment function; (2) conversion ofthe charge density grid to reciprocal space by means of a three-dimensional fast Fourier transform (3D-FFT); (3) solution for thereciprocal-space electrostatic potential via multiplication by anoptimized influence function; (4) conversion of the reciprocal-space potential grid to real space by means of a 3D-FFT; (5)evaluation of the electrostatic field on the grid by means of afinite-difference operator; (6) interpolation of the field at the loca-tion of the atomic partial charges by means of the same assignmentfunction used in the first step. In the ik-differentiation variant,87,88

the fifth and sixth steps are replaced by: (5) evaluation of thereciprocal-space field through multiplication by ik; (6) conversionof the reciprocal-space field grid to real space by means of three3D-FFTs, one for each field component.

The influence function describes the electrostatic potential gen-erated at the different grid points by a unit (� � b)-charge at theorigin. It is stored in the form of its corresponding value at each ofthe reciprocal-space grid points. A key to the accuracy of thealgorithm is to preoptimize this function so that it compensates forerrors inherent to the discretization process, the use of an approx-imate assignment function, and the use of an approximate finite-difference operator.84 When the virial is to be calculated or whenthe box dimensions may vary in the course of a simulation, sixgrids are computed simultaneously, containing the relevant deriv-atives of the optimal influence function with respect to the boxparameters.13 The optimization of the influence function (and theevaluation of its derivatives when required) is computationallyexpensive. In simulations without variations of the box parameters,however, this function is constant (as well as its derivatives), andthe calculation needs to be performed only once at the beginningof a simulation. In simulations involving a variation of the boxparameters, the accuracy of the influence function may progres-

sively deteriorate with time as the box changes shape and size.Two mechanisms are then used to improve the accuracy of thecurrent influence function at reasonable computational costs. First,the derivative information computed together with the optimizedinfluence function is used to apply a first-order correction to thecurrent influence function upon variation of the box parameters.13

Second, the accuracy of the algorithm88 may be reevaluated atperiodic intervals, and a reoptimization of the influence function(and recalculation of its derivatives) undertaken when this accu-racy falls below a user-specified threshold.

In the following paragraphs, the P3M algorithm is described inmore details.

A general triclinic box may be discretized by means of a gridG, defined by the number of subdivisions Na, Nb, and Nc alongthe a, b, and c box-edge vectors. It will be convenient to introducethe diagonal matrix N� with elements Na, Nb, and Nc. The matrixH� is then defined as

H� �Na�1ax Nb

�1bx Nc�1cx

Na�1ay Nb

�1by Nc�1cy

Na�1az Nb

�1bz Nc�1cz

� L� N� �1. (61)

The volume of a grid cell is noted VG � �H� �.Each point of the real-space grid G is associated with an index

n � (na, nb, nc), with na � [0; Na � 1], nb � [0; Nb � 1],and nc � [0; Nc � 1]. Points outside this range are periodiccopies of points within the range. The real-space vector corre-sponding to a grid point n may be written in the different repre-sentations:

�n N� �1n and rn H� n. (62)

Similarly, each point of the reciprocal-space grid G is associatedwith an index l � (la, lb, lc), with la � [�Na/ 2 � 1; Na/ 2],nb � [�Nb/ 2 � 1; Nb/ 2], and nc � [�Nc/ 2 � 1; Nc/ 2]. Pointsoutside this range are periodic copies of points within the range.The reciprocal-space vector corresponding to a reciprocal-spacegrid point n may be written in the different representations as

�l 2�l and kl 2�tL� �1l. (63)

All gridded functions, that is, real- or reciprocal-space functionsthat only take a value at a grid point of G, will be indicated witha “g” subscript.

The forward 3D-FFT operation converts a gridded functionfg(rn) on the grid G into its finite Fourier coefficients fg(kl) on thesame grid, as

fg�kl� VG �n�G

fg�rn�e�ikl�rn. (64)

The backward 3D-FFT performs the reverse operation, namely,

fg�rn� V�1 �l�G

fg�kl�eikl�rn. (65)


The P3M algorithm starts by distributing the Nq atomic partialcharges qi at locations ri within the computational box onto theneighboring grid points (taking periodicity into account), so as togenerate the charge-density grid sg. This assignment is performedas

sg�rn� �i�1

Nq

qi�g�rn; ri� (66)

with

�g�rn; r� P�rn r�, (67)

where P is a so-called assignment function (discussed in moredetails in the next subsection). The charge density grid is thenconverted to its (complex) reciprocal-space representation sg(kl)by applying a forward 3D-FFT to sg(rn).

The reciprocal-space potential, that is, the potential generatedby the corresponding gridded (� � b)-charges is then computed inreciprocal space as

��,g�kl� �o�1Gg

†�kl�sg�kl�, (68)

where Gg† represents the Fourier coefficients of the influence

function. If all charges were located exactly at grid points or if thegrid spacing was infinitesimal, the quantity Gg

† would be given bykl

�2�(akl). However, in practice, a significant gain in accuracy isreached by optimizing Gg

† to compensate for errors linked with thediscretization procedure, taking into account possible variations inthe shape and size of the computational box. To this purpose, theinfluence function Gg

† is defined as13

Gg†�kl� Gg

o�kl� Tr��go�kl��

tL� o��1�tL� tL� o�, (69)

where Ggo is the influence function optimized for a given set L� o of

box parameters and ��go contains the corresponding first-order de-

rivative information in the form

��

go�kl� �

�Ggo�kl�

�L� otL� o. (70)

The calculation of the quantities Ggo and �g

o is described in moredetails in the next subsection. The optimal influence function Gg

o isonly optimal for a specific set L� o of box parameters L� . When theshape and size of the computational box may vary, the second termin eq. (69) includes a first-order correction to the influence functionoptimized at L� o, based on the derivative information calculatedsimultaneously.

The reciprocal-space contribution �� to the total electrostaticenergy is given by

�� 2�oV��1 �i�1

Nq �j�1

Nq �l�G,l�0

qi�g�kl; ri�qj�*g�kl; rj�Gg†�kl�. (71)

For computational efficiency, this pairwise sum is evaluated as asingle sum, through

�� 2�oV��1 �l�G,l�0

Gg†�kl��sg�kl��2. (72)

The (approximate) forces associated with the energy contribu-tion �� are obtained through the evaluation of the gridded field Eg

and its interpolation at the location of the charges. The reciprocal-space force on atom i is then written as

F�,i qiE�ri� (73)

with

E�r� VG �n�G

P�r rn�Eg�rn�. (74)

The same assignment function P should be used here and for thecharge assignment, to ensure conservation of the total linear mo-mentum during the dynamics.84 The gridded field Eg to be used ineq. (74) can be obtained in either of two ways.

The first method (finite-difference) relies on performing a back-ward 3D-FFT of the potential to obtain the (real) real-space po-tential, and using a finite-difference approximation to compute thegridded field as

Eg�rn� � �n��G

iVGDg�rn rn��,g�rn��, (75)

where iVGDg is a so-called finite-difference operator (discussed inmore details in the next subsection).

The second method (ik-differentiation) relies on computing theexact gridded field in reciprocal-space as88,89

Eg�kl� �ikl��,g�kl�. (76)

One then performs three backward 3D-FFTs (one for each Carte-sian component) to obtain the corresponding (real) quantity Eg inreal-space.

Quantities involved in the P3M algorithm. The assignment func-tion P of order p [eqs. (67) and (74)] performs the distribution(interpolation) of a continuous function at an arbitrary locationonto (from) values at the neighboring p3 grid points. This functionis defined as56

P�r� VG�1 �

n��3

P�r � L� n� (77)

with

P�r� wp��H� �1ra�wp��H� �1rb�wp��H� �1rc�. (78)

Here, wp(�) is a normalized one-dimensional function vanishingfor �� p/ 2. These functions are listed in Table 7. The assign-


ment scheme is formulated in terms of oblique coordinates. Thus,in the general case, the distribution (interpolation) of the functionfrom (at) an arbitrary location onto (from) the neighboring gridpoints is not necessarily correlated with the Cartesian distancebetween the points (this is only the case for a rectangular compu-tational box).

The Fourier coefficients P of the assignment function P aregiven by

P�k� wp��tH� ka�wp��

tH� kb�wp��tH� kc�, (79)

where wp(�) is the continuous Fourier transform of wp(�), whichevaluates to

wp�� 2��1sin��/2�p�1 �� . (80)

The finite-difference operator iVGDg of order q estimates thegradient of a given function at a grid point based on the values ofthe function at the 6q neighboring grid points along the three boxaxes. The operator Dg is defined as56

Dg�rn� �j�1

q

cjdg, j�rn�. (81)

In this expression, the centered-difference operator iVGdg, j gen-erates the gradient contribution evaluated from the function at thesix neighboring points distant from the grid point by �jNa

�1a,�jNb

�1b, and �jNc�1c along the three box axes. The operator dg, j

is given by

dg, j�rn� R� S�eg, j�rn�, (82)

where R� and S� are the matrices defined in eqs. (4) and (5), eg, j isthe corresponding operator in terms of oblique fractional coordi-nates, that is,

eg, j,a�rn� iVG�1�2jNa

�1a��1�nb�nc �m��

��na�j�Nam �na�j�Nam�,

(83)

with similar expressions for the b and c components.The Fourier coefficients Dg of the operator Dg are given by

Dg�kl� �j�1

q

cjdg, j�kl�, (84)

where

dg, j�kl� R� tS� �1eg, j�kl� (85)

and

eg, j,a�kl� � jNa�1a��1sin� j�tH� kla�, (86)

with similar expressions for the b and c components. Takentogether, eqs. (84), (85), and (86) may be written

Dg�kl� tH� �1 �j�1

q

cjj�1�sin( j[tH� kl]a)

sin( j[tH� kl]b)sin( j[tH� kl]c)

�. (87)

Note that the exact difference operator (as used in ik-differentia-tion) corresponds to

Dg�kl� kl. (88)

For a given finite-difference operator of order q, the coefficients cj

in eq. (81) may be selected so as to minimize the differencebetween the corresponding finite-difference operator and the ex-act-difference operator. In reciprocal space, this leads to the re-quirement

tH� �1 �j�1

q

cjj�1�sin(2�jNa

�1la)sin(2�jNb

�1lb)sin(2�jNc

�1lc)� � kl. (89)

Solving the resulting system of equations results in the optimalcoefficients listed in Table 8.

Table 7. One-Dimensional Functions wp(�) Used to Define the Assignment Functions [eqs. (77)and (78)].

p wp(�) Range

1 1 �� 1/22 1 � �� 13 �(3/4)(4�2 � 1) �� 1/2

(1/8)(2�� 3)2 1/2 � �� 3/24 (1/6)(3��3 � 6�2 � 4) �� 1

�(1/6)(�� 2)3 1 � �� 25 (1/192)(48�4 � 120�2 � 115) �� 1/2

�(1/96)(16�4 � 80��3 � 120�2 � 20�� 55) 1/2 � �� 3/2(1/384)(2�� 5)4 3/2 � �� 5/2

The functions wp(�) vanish for �� p/ 2.


For a given set of box parameters characterized by the matrixL� o, the influence function that optimally compensate for discreti-zation errors can be computed as84

Ggo�kl�

Dg�kl� � �¥m��3 kl,mkl,m�2��akl,m�P2�kl,m�

Dg2�kl��¥m��3 P2�kl,m�2 , (90)

where

kl,m kl�N� m kl � 2�tH� �1m (91)

is an alias vector of k (with m � �3). Equation (90) is valid for l� G and l � 0, together with Gg(0) � 0. In practice, thesummation over alias vectors is restricted to m vectors with integercomponents in the range [�mmax . . . mmax]. A value of 2 or 3 formmax is usually sufficient to reach convergence. The quantity P isgiven in by eq. (79), the quantity Dg by eqs. (87) (finite-difference)or (88) (ik-differentiation), and the quantity � by eq. (43) (seeTable 4). The negative derivative of the optimal influence functionwith respect to the box parameters L� o, amplified on the right bytL� o may be calculated simultaneously with the influence functionas

�� go�kl�

1

Dg2�kl��¥m��3 P2�kl,m�2 �

m��2

P2�kl,m�

kl,m2 kl,m � Dg(kl)

� Dg(kl) � kl,m 2[kl,m � Dg(kl)]kl,m � kl,m

kl,m2

2[kl,m � Dg(kl)]Dg(kl) � Dg(kl)

Dg2(kl)

��(akm)

�[kl,m � Dg(kl)]kl,m � kl,m

kl,m2 akl,m��(akl,m)� (92)

with ��(�) � d�(�)/d� valid for l � G and l � 0, together with��g

o � 0� .

Replica-Exchange Simulation

To obtain canonical distributions for complex molecular systems,efficient sampling of the configurational space is necessary. Find-ing the global minimum on the typically rough potential energylandscape of a peptide or protein is likewise difficult. In recentyears, the replica-exchange method90–95 (also known as parallel

tempering95) has received much attention. A number of noninter-acting replicas are simulated simultaneously at different conditions(e.g., different temperatures). After a given simulation time, anexchange between two replicas is attempted, followed by another(individual) simulation period. The method has been applied tobiomolecular systems,96–99 using Monte Carlo (MC) techniquesand molecular dynamics (REMD) to propagate the individualreplicas. The probability of each state x � (r, p) in the canonicalensemble at temperature T is proportional to the weight factor

W� x� exp��H�r, p��, (93)

where H is the Hamiltonian and � � 1/kBT, kB being Boltz-mann’s constant. The weight factor for the global state X deter-mined by the states of the M replicas is the product of the singleweights, that is,

WREM�X� exp��i�1

M

�iH(ri, pi)�. (94)

After a fixed number of MD integration steps, an MC exchangebetween two replicas is attempted (changing from state X to stateX�). To sample canonical ensembles at each temperature, thedetailed balance condition on the transition probability w(X3 X�)

WREM�X�w�X 3 X�� WREM�X��w�X� 3 X� (95)

has to be fulfilled. This can be satisfied, for instance, by the usualMetropolis criterion

p�X 3 X�� w�X 3 X��

w�X� 3 X� 1 for � � 0,

exp(��) for � � 0 , (96)

with

� ��i �j��U�rj� U�ri��, (97)

where U(r) is the potential energy associated with the configura-tion r. If the exchange was successful, the momenta of the ex-changed replicas are scaled to correspond to their new tempera-tures.

An extension of the replica-exchange method to sample theisothermal–isobaric ensemble has been suggested.100 In this case,an additional term incorporating the pressure and volume changeappears in the exchange probability

� ��i �j��U�rj� U�ri�� iPi �jPj��Vj Vi�. (98)

Unfortunately, the application of the REMD method to explicitsolvent simulations, although successful for small systems, israther difficult.101–107 Because the exchange probability betweentwo states decreases with increasing system size, explicit-solventsimulation requires many more states (replicas) separated by smalltemperature differences. This problem may be alleviated by notexchanging a thermodynamic property like the temperature be-tween the individual replica, simulations but rather altering spe-

Table 8. Optimal Weighting Coefficients cj ( j � q) for the Finite-Difference Operator.

q c1 c2 c3 c4 c5

1 12 4/3 �1/33 3/2 �3/5 1/104 8/5 �4/5 8/35 �1/355 5/3 �20/21 5/14 �5/63 1/126


cific interactions.108–110 Then, the replicas are distinguished bytheir Hamiltonians

Hi�ri, pi� K�pi� � Ui�ri� (99)

using for each simulation a different potential energy function Ui.In MD�� the different Hamiltonians Hi are defined using acoupling parameter � and a perturbation topology. The replica with�i � 0 corresponds to state A (normal topology) in a perturbationsimulation, the replica at �i � 1 to state B (fully perturbed state;with, e.g., scaled-down nonbonded or bonded interaction terms).The other replicas are distributed in between these two (0 � �i �1).

Inserting the individual Hi into eq. (94), leads to

WREM�X� exp��i�1

M

�iHi(ri, pi)�, (100)

and, using the detailed balance criterion to

p�X 3 X�� w�X 3 X��

w�X� 3 X� 1 for � � 0,

exp(��) for � � 0, (101)

with

� �i�Ui�rj� Ui�ri�� j�Uj�rj� Uj�ri��, (102)

where the potential energy of the two configurations ri and rj

needs to be evaluated twice, with Hamiltonians Hi and Hj.The replica-exchange method was implemented in MD��

based on sockets and tcp as communication protocol. A serverdistributes the short MD runs corresponding to the different rep-licas to a (dynamical) number of clients. After a given number ofsimulation steps has been carried out, the client reports back to theserver the final (potential) energies (evaluated using the Hamilto-nian Hi and the Hamiltonian Hj (if different), and the final con-figuration. The server then calculates the switching probability p(i3 j), draws a random number and, if the switch is successful,exchanges the states. As soon as a client is free, the next replicagets assigned to it.

Replicas can differ in the temperature and in the couplingparameter �. A replica-exchange state consists of replicas for allpossible � values at each temperature Ti (M � N� � NT replicas).The MC exchange attempt is alternated between exchanges of(neighboring) �’s and temperatures T.

Coarse-Grained Simulation

Most molecular simulations are making use of atom-level (AL)models. This limits the time scale of such simulations for solvatedmacromolecules to the nanosecond range. Longer time scales canbe reached by treating molecules or molecular fragments as singleparticles or beads, whose motion is simulated using a simple forcefield describing interbead interactions. When the energy functionof such a coarse-grained (CG) model is chosen to be smooth andshort-ranged, the efficiency of CG simulations can be orders of

magnitude (103–105) higher than the corresponding AL simula-tions, be it at the expense of the loss of atomic detail and someaccuracy.111–115

A recently proposed CG model116 for liquid simulations has thesame functional form as the GROMOS force field,3,117 except forthe use of a switching function118 for the nonbonded Lennard–Jones and electrostatic interactions at distances just below thecutoff distance. This CG model was implemented into GRO-MOS05 (MD�� only), however with a slightly different switch-ing function, because the GROMACS one118 appeared to be dis-continuous and led to nonconservation of energy in MDsimulation.

In the absence of switching, the nonbonded interaction energybetween particles i and j can be written as (rij � ri � rj)

V�rij� ��1,6,12

V��rij� ��1,6,12

c��rij� (103)

with (r � �r�)

��r� r�� (104)

and

c1 qiqj

4��0�1

c6 �4�ij�ij6 �C6�i, j�

c12 �4�ij�ij12 �C12�i, j�, (105)

where standard notations for atomic charges (qi) and van derWaals interaction parameters (C6 and C12) have been used. In theCG model all nonbonded interactions are smoothly switched tozero over the range [Rsw, Rc], where Rsw denotes the start of theswitching and Rc the cutoff radius. In this model116 one has Rc �1.2 nm and Rsw � 0 nm for the Coulomb interaction and Rsw �0.9 nm for the van der Waals interactions. The nonbonded inter-action energy function including switching reads for the threeterms in eq. (103)

��s �r� ��(r) r � Rsw

��(r) � S�(r) Rsw � r � Rc

0 r � Rc

. (106)

Requiring that the functions S�(r) switch the energy, the force,and the derivative of the force smoothly (without discontinuities)to zero at r � Rc, yields the conditions

S��Rsw� S��Rsw� S��Rsw� 0 (107)

and

��s �Rc� ��

s��Rc� ��s��Rc� 0. (108)

The conditions of eq. (107) are satisfied by a fourth-degree poly-nomial


S��r� �1

3A�r Rsw�3

1

4B�r Rsw�4 C. (109)

The conditions of eq. (108) determine the constants

A �� 1� Rsw �� 4�Rc

Rc��2�Rc Rsw�2 , (110)

B �� 1�Rsw �� 3�Rc

Rc��2�Rc Rsw�3 , (111)

C 1

Rc�

1

3A�Rc Rsw�3

1

4B�Rc Rsw�4. (112)

The expression for the shifted or switched force on particle i byparticle j for the three nonbonded interaction terms V�

s (rij) is then

f�is �rij� �

�V�s �rij�

�rij

�rij

�ri �c��

s��rij�rij

rij, (113)

with

��s��r� ��(r) r � Rsw

��(r) � S��(r) Rsw � r � Rc

0 r � Rc

, (114)

��r� ��

r��1 , (115)

and

S��r� �A�r Rsw�2 B�r Rsw�3. (116)

We note that the switched potential energy determined by eq.(106) and forces from eq. (113) are only correct for a distancedependence of the potential energy function of the form of eq.(104). So, it cannot be used in the presented form if the soft-coreinteraction or reaction field forces as defined in GROMOS are tobe used.

Free-Energy through One-Step Perturbation

The calculation of relative binding free energies of many ligands toa common receptor is of relevance for drug design and screeningpurposes, and for obtaining a better understanding of interactionsgoverning molecular complexation in general. The one-step per-turbation technique119 allows for the calculation of a great manyrelative free energies from a single simulation of a (not necessarilyphysically meaningful) reference state.120–127 The idea behind themethod is to simulate a judiciously chosen reference compound Rgenerating an ensemble of structures that contains conformationsrepresentative for many physically relevant compounds. The freeenergy difference between any real ligand A and the referencecompound R can be obtained from the perturbation formula

�GAR �GA �GR �kBT ln�e��HA�HR�/kBT�R, (117)

where the angular brackets indicate the ensemble average of theconfigurations generated in a simulation of R. HA and HR are theHamiltonians for the real compound ( A) and the reference com-pound (R), respectively. Because this expression involves thedifference between two Hamiltonians, only interactions that differbetween compounds A and R need to be reevaluated over theensemble. This allows for the calculation of thousands125 to mil-lions126 of relative free energies from a handful of simulations ofreference states R.

The success of the method critically depends on the choice ofthe reference state R; it should allow wide sampling, but not sowide that insufficient statistics is obtained. One of the key elementsthat allow wide sampling is the use of soft-core nonbonded inter-actions, which allows for a spatial overlap between these atoms. InGROMOS96, the soft-core nonbonded interaction was chosen tobe of the form3,4,119

Vsc�rij� 4�ij�ij

6

sLJ�i, j��2�ij6 � rij

6 � �ij6

sLJ(i, j)�2�ij6 � rij

6 1�

�qiqj

4��0�1� 1

(sC(i, j)�2 � rij2)1/ 2

1

2Crfrij

2

(sC(i, j)�2 � Rrf2 )3/ 2

1 1

2Crf

Rrf.

(118)

In GROMOS96, the soft-core parameters sLJ and sC were takenequal for all soft-core atom pairs. In GROMOS05 (MD�� only),sLJ(i, j) and sC(i, j) are calculated by combining the distinctsoftness parameter specified per atom, allowing fine tuning of thereference state, that is,

s�i, j� 1

2(s(i) � s( j)) s(i) � 0, s( j) � 0

s(i) s(i) � 0, s( j) 0s( j) s(i) 0, s( j) � 00 s(i) 0, s( j) 0.

(119)

The GROMOS�� postprocessing programs pt_top (to gen-erate a real topology from a topology and a perturbation topology)and ener (to recalculate the interaction energy of specified atoms)or m_pt_top and m_ener (to do the same for multiple physicalcompounds at the same time), and dg_ener (to calculate therelative free energies) may be used to analyze the reference stateensemble.

Features under Construction

A number of algorithmic and force field developments are cur-rently investigated and only provisionally implemented in GRO-MOS05. These are not yet part of the standard software.

Polarizability

During the past decade, the accuracy of the GROMOS force fieldcould still be improved by reparametrizing it to reproduce thethermodynamic properties of small molecules.117,128,129 The latestparameter set, 53A6, reproduces the free energies of apolar (cy-


clohexane) and of polar (water) solvation for typical biomoleculesto within 1–2 kJ mol�1 (about kBT/ 2).117 However, this studyindicated that the lack of explicit electronic polarizability is lim-iting a further increase in the accuracy of nonpolarizable forcefields. Various possibilities to explicitly account for polarizabilityin MD simulations have been reviewed.130 In our view, the mostpromising approach in terms of striking a balance between accu-racy on the one hand and simplicity and computational efficiencyon the other is the so-called charge-on-spring (COS) type ofmodels.131 A COS model for polarizable water that is, in principle,compatible with the GROMOS type of force fields has beendeveloped132,133 and a GROMOS software version with COSpolarization is in its test phase, based on the implementationpreviously sketched.130

Enhanced One-Step Perturbation

The efficiency of the one-step perturbation technique can be en-hanced along two lines. First, because the choice of the nonphys-ical reference state R is completely free, even nonatomic referencecompounds can be used.127 Using the concept of soft-core atoms,one may use atoms of dual character in the reference compound:atom i may interact through a soft-core interaction with atom j, butthrough a normal interaction with atom k. In this manner the partof the system that is softened to enhance the sampling can berestricted such that no unnecessary softness is introduced thataggravates the sampling. Second, when evaluating the differenceHA � HR in eq. (117), the Hamiltonian HA(rR) or energy of thereal compound A is to be calculated using the ensemble rR ofconfigurations of the reference compound R in the trajectory of R.This requires a definition of the interaction sites of HA in terms ofatom (mass) positions of HR, which should be the same for allconfigurations of the trajectory of R. However, because eq. (117)is independent of translation and rotation of the coordinate systemthat is used to define HA, additional sampling in the form oftranslation and rotation of the real compound with respect to thetrajectory configurations of the reference compound R can becarried out,122 for example, through limited translation and rota-tion. Furthermore, if the reference compound is composed ofatomic soft sites on each of which more than one real ligand atomcan be superimposed, the calculation of HA � HR for the manycombinations of real ligand atoms at the different soft sites can bedecomposed in single soft-site calculations, of which the contri-butions to HA � HR can be added. This allows a considerableincrease in efficiency.

Code Organization, Implementation

MD Engine in FORTRAN: PROMD

The FORTRAN MD engine (PROMD) is an enhancement of theGROMOS96 MD engine. It is written in FORTRAN77 except forthe use of included files and macro preprocessing. Macros are inparticular used to get rid of unnecessary features (such as four-dimensional simulation, unused periodicity code, or unused per-turbation code) so as to improve the performance for specificapplications through the use of a specialized code. To facilitate

performance tuning, timing routines have been included, and thetime spent within various components of the program is reported atthe end of each simulation. Major additional algorithmic features(with respect to GROMOS96) have been described (see earlier).

MD Engine in C��: MD��

The C�� MD engine (MD��) has been written from scratch.The major motivation was to further increase the modularity andtherefore the extendability of the MD program. The code is splitinto two parts, the first one being an MD library containing basicfunctions necessary to run an MD simulation, the second one beingthe actual MD program. This second part is very small. It istherefore easy to write other specialized MD programs that makeuse of a subset of the functions provided in the library or applythem in a different order. The source code of the library is, in turn,split up into nine different parts: math, simulation, topology, con-figuration, algorithm, interaction, io, util, and check (representedas C�� namespaces).

● math contains classes for vectors, matrices, and vector arrays,mathematical operations, physical constants, and periodicboundary treatment.

● simulation contains the simulation parameters supplied to run anMD or SD simulation or an EM.

● topology contains the topology of the simulated system, possiblyalso including a perturbation topology.

● configuration contains the state of a system: its coordinates,velocities, forces, restraints data, and so on.

● algorithm contains classes that use information from simulationand topology to act upon a configuration. All steps during anMD or SD simulation or EM can be carried out using analgorithm.

● interaction contains the largest algorithm: the energy, forces,and virial evaluation. Here, all interaction terms and their pa-rameters are defined. Because of its size, interaction is a separatepart, although it formally belongs to algorithm. The interactionpart is further split into bonded, nonbonded, and special inter-actions.

● io contains classes to read in or write out information. All fileaccess is block oriented and the files are human readable.

● util contains a few extra classes that are necessary to set up asimulation but which do not exactly belong to it. Parsing ofcommand line arguments, generation of initial velocities, orsetting of debug levels are examples of classes found herein.

● check contains test routines. Testing includes the automaticcalculation of energies under different conditions as well as thecalculation of forces, virial tensor, and energy �-derivatives andtheir comparison to values obtained by finite difference calcu-lations.

One step of an MD or SD simulation or EM consists of severalAlgorithms (Fig. 3) applied to the Configuration in theright order. The Algorithm_Sequence class (Fig. 4) is acontainer for all these algorithms. When a simulation is set up,they are inserted in the correct order into the Algorithm_Se-quence. Before the start of a simulation, all algorithms will beinitialized (by calling the init ( ) function). During an MD


step (Algorithm_Sequence::run( )), the algorithms areapplied (by calling Algorithm::apply( )). The forcefield itself is also an algorithm, which, when applied, calcu-lates the energies, forces, and virial contribution of all force fieldterms for the complete system. The force field terms themselvesare Interaction classes. The Forcefield is therefore a con-tainer to store the different Interaction objects (in analogy tothe Algorithm_Sequence and Algorithm classes). Whenthe force field is applied, it calls calculate_interac-tions( ) on all interaction objects. There are distinct interac-tion objects for the covalent interactions (bond length, bond angle,improper–dihedral, and torsional–dihedral interactions), the non-bonded interactions (pairlist construction, long-range interactions,and short-range interactions) and the nonphysical interactions(atom position, atom distance, dihedral angle, NOE, or J-valuerestraints). It is very easy to add a custom Interaction class tocalculate a nonstandard interaction.

The classes corresponding to the steps in the MD, SD, or EMalgorithm are shown in Table 9 and an overview of the (non-bonded) interaction classes is given in Figure 5. The Nonbond-ed_Sets contain independent subsets of the nonbonded interac-tions. Their calculate_interactions ( ) method may becalled in parallel (using either shared or distributed memoryparallelization). The Nonbonded_Sets share (through theNonbonded_Interaction) a pairlist construction algorithm,which they call to create the part of the complete pairlist relevantto them. These different parts of the pairlist stay together with theNonbonded_Set and need never be assembled into the completepairlist. To gain flexibility, the calculation of the individual atom–atom pair interaction is further split up into a Nonbonded-_Outerloop (loops over the atom–atom pairs), a Nonbond-ed_Innerloop (prepares the parameters necessary to calculatethe interaction), and a Nonbonded_Term (calculates the atom–atom pair interaction energy, force, and virial contribution). TheStorage class provides directly accessible (local) memory foreach Nonbonded_Set.

Efficiency

The main goal for writing a new C�� MD engine was to furtherimprove on modularity (using some object-oriented features) andextendability (using clear and common interfaces between themodules). Nevertheless, a simulation code has to be reasonablyefficient to be of practical use. The complete code is written instandard C��,134 no language extensions or machine-specificparts are used anywhere, resulting in a highly portable program.This means that the compiler has to do all machine-specific opti-mizations. We believe that the absence of any machine-specificparts of code, which require duplication to be able to run ondifferent machines, facilitates future modification. Furthermore,current compilers are getting ever better at producing fast pro-grams, making use of the specific features available on the ma-chine.

In the inner loops of the interaction calculation, templatesare used to generate specialized code. There are, for instance,specialized periodicity classes for the different implemented typesof periodic boundary conditions (vacuum, rectangular, truncatedoctahedral, and triclinic). The Innerloop methods are calledwith the boundary type as a template argument. Thus, the compilerwill generate different specialized versions of the inner loops fordifferent boundary conditions automatically. In the same manner,the interaction function term of the nonbonded interaction can alsobe chosen (e.g., with or without switching function for nonbondedinteractions) without any if statement required in the compiledinner loop. An example code fragment is shown in Figures 6 and7. The same technique is used to implement perturbation simula-tions and different definitions of the virial tensor.

Some algorithms do rely on information from the previousintegration step. To help implementing those kind of algorithms,the complete current and old state (positions, velocities, forces,energies, restraint and constraint data, averages, and so on) of thesimulation are stored. During the leap-frog algorithm, the currentstate becomes the old state and the updated information is stored inthe new current state. This transfer is done by a simple (and fast)pointer exchange. This duplication slightly increases memory us-age (but the required space is still small compared to that used tostore the pairlists).

A comparison of the efficiency of the C�� code with respectto the GROMOS96 MD engine (in FORTRAN) is given in TableFigure 3. Interface of the Algorithm class.

Figure 4. Interface of the Algorithm_Sequence class.


2. This comparison shows that MD��, using the standard pairlistalgorithm is approximately a factor 2 slower than GROMOS96(standard pairlist algorithm); improved algorithms (like a grid-based pairlist construction) may have a large impact on the per-formance reducing the time spent to two-thirds for the membraneand even by a factor of 3 for the protein system. Note that theoptimized pairlist construction algorithm implemented in the FOR-TRAN MD engine (PROMD) only benefits from more efficientprocessor cache usage but still scales as O(N2) with system size.Still, for the systems tested here, it achieves equal overall effi-ciency as the O(N) scaling grid-based pairlist algorithm inMD��. The future will show whether the improved extendabilityof MD�� will, through improved algorithms, lead to a fasterC�� code than the FORTRAN (PROMD) code, or whetherslightly inferior performance is the price to pay for a more struc-tured code layout.

Debugging Information

It is often difficult to figure out what is going on during an MD orSD simulation or an EM, and many users tend to use the programas a black box. MD�� tries to improve this situation by enablingthe user to select a tunable amount of information to be printed outduring the simulation. Every (output or debugging) message isassociated with a debugging level, and the message is printed onlyif the requested debugging level is high enough. Additionally,every code section belongs to a module and a submodule. Differentdebug levels can be specified for all combinations of modules andsubmodules. In that way, fine-grained control is achieved on how

much information from which part of the MD�� code should beprinted.

Parallelization

Computationally, the interaction calculation is by far the mostexpensive part of an MD or SD simulation or an EM, while thenonbonded interactions constitute the bulk of the effort. Again,MD�� is focused on achieving parallelization without compli-cating the code. The nonbonded interaction is split up into Non-bonded_Sets, each containing its own storage space for apairlist, energies, forces, and virials. In this way, the standard codeis ready for shared and distributed memory parallelization withoutany need for code duplication. If the system is using distributedmemory, the (updated) positions have to be copied from the masterto all other processes before the next interaction calculation. Whilecomposing the pairlist in parallel, only a subset of atoms isconsidered per process, so that each processor creates its ownpartial and local pairlist. The interactions are calculated from thispartial pairlist and stored in local arrays. This ensures synchroni-zation for shared memory machines and replicated data parallel-ization for distributed memory systems. After the partial interac-tion calculations have finished, the energies, forces, and virials ofall nonbonded sets are summed up and stored in the Configu-ration of the master process.

MD�� can use OpenMP (www.openmp.org) for shared mem-ory and MPI (www.mpi-forum.org) for distributed memory paral-lelization. Reasonable parallelization (using a small number ofparallel processes) can be achieved with only a few additional lines

Table 9. Classes Corresponding to MD Algorithm Steps.

1. Write position and velocity components. Out_Trajectory2. Remove center of mass motion. Remove_COM_Motion3. Calculate (unconstrained) forces and energies from the potential energy function

(using nearest image convention in case of periodic boundary conditions).ForcefieldBond_InteractionAngle_InteractionImproper_Dihedral_Interaction

Save these. Dihedral_InteractionNonbonded_InteractionPosition_Restraints_InteractionDistance_Restraints_InteractionNOE_Restraints_InteractionJValue_Restraints_Interaction

4. Satisfy position constraints. Position_Constraints_Interaction5. Update the velocities using the leapfrog scheme. Leapfrog_Velocities6. Apply temperature coupling [weak coupling or Nose–Hoover(-chains)]. Berendsen_Thermostat

Nose_Hoover_Thermostat7. Update the positions using the leapfrog scheme. Leapfrog_Positions8. Satisfy distance constraints (using SHAKE, M-SHAKE, or LINCS). Shake

MShakeLincs

9. Calculate temperature(s). Temperature_Calculation10. Calculate pressure. Pressure_Calculation11. Apply pressure scaling (weak coupling). Berendsen_Barostat12. Update lambda and topology for slow-growth simulations. Slow_Growth13. Calculate total energies, averages, and fluctuations. Energy_Calculation

Save these.


of code (almost) completely separate from the nonbonded routines(see Table 2).

Analysis Modules: GROMOS��

All the FORTRAN analysis programs of GROMOS96 have beenrewritten in C��. They accept a standard set of command linearguments to specify input. It is easy to add new analysis programsusing the functionality provided within the GROMOS�� library.Following is a short description of the existing programs.

Setup of Simulations (Preprocessing)

● make_top builds a topology from a building block sequence.● com_top combines two topologies.● con_top converts topologies to a different force field param-

eter set.

● red_top reduces topologies by specified parts.● pt_top combines topologies with perturbation topologies to

produce new topologies or perturbation topologies.● pert_top creates a perturbation topology to perturb specified

atoms to dummies.● check_top checks topologies for common mistakes.● pdb2g96 converts a pdb (Protein Data Bank) structure into

GROMOS coordinates.● build_box builds a simulation box containing N molecules at

a specified density.● ran_box builds a simulation box containing N molecules at a

specified density, placing and orienting them randomly.● bin_box builds a simulation box containing a binary mixture

at a specified density.● sim_box puts a simulation box around a molecule and fills it

with solvent molecules from an equilibrated solvent configura-tion.

Figure 5. Illustration of the Interaction classes in MD��. The red arrows denote a is-a relation-ship, the black arrows has-a. All Interaction classes inherit from Interaction and, therefore, canbe stored in the Force field, which is a vector of Interaction classes. The Nonbonded_I-nteraction consists of a Pairlist_Algorithm (either a Standard_Pairlist_Algorithmor a Grid_Pairlist_Algorithm) and (depending on parallelization) one or more Nonbonded-_Sets. Those, in turn, consist of Storage (to locally store forces, energies, virial tensor, and pair lists)and an Outerloop (to calculate the interactions). The Outerloop relies on the Innerloop and onTerm to calculate the interactions. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com.]


● ran_solvation builds a simulation box around a moleculeand fills it randomly with solvent molecules.

● check_box checks box properties (size).● copy_box multiplies a box in any direction.● explode increases intermolecule distances to vacuum condi-

tions.● cry applies rotations and translations to a system to create a

crystal unit cell.● ion replaces a specified number of solvent molecules by ions.● gch generates hydrogen atom coordinates for a molecule.● gca generates atomic Cartesian coordinates from a set of inter-

nal coordinates.● mk_script prepares an MD, SD, or EM job script.

Analysis of Trajectories (Postprocessing)

● tstrip removes solvent from a trajectory.● filter filters out specified atoms from a trajectory.

● cog calculates center of geometries for specified atoms.● tser calculates time series of specified properties (distances,

angles, torsions, order parameters, etc.).● tcf calculates time correlation functions of time series.● dist calculates distributions of specified properties.● ditrans monitors dihedral-angle transitions.● propertyrmsd calculates root-mean-square differences for a

set of properties.● dipole calculates dipole moments with respect to the center of

molecules.● rmsd calculates atom-positional root-mean-square differences

between structures.● rmsf calculates atom-positional root-mean-square fluctuations

for specified atoms.● ene_ana calculates averages, fluctuations, and error estimates

for energies, pressure, and volume.● epsilon calculates the dielectric permittivity for liquids.● visco calculates the shear viscosity of liquids.● rgyr calculates the radius of gyration.● rdf calculates the radial distribution function for selected at-

oms.● mdf gives the time series of the closest particle index to a

selected atom.● m_widom performs widom particle insertion.● sasa calculates the solvent accessible surface area (SASA) for

a specified part of a molecule.● hbond analyses hydrogen bonding.● dssp analyses secondary structure elements.● prep_noe prepares for an NOE calculation.● noe calculates NOE distances.● post_noe analyses NOE distances.● oparam calculates order parameters for lipids in membranes.● nhoparam calculates N—H order parameters.● diffus calculates the diffusion coefficient of specified atoms.● rmsdmat calculates the RMSD between all structure pairs in a

trajectory.● cluster analyses an RMSD matrix to separate the structures

into clusters.● postcluster analyses the cluster output for lifetimes, fold-

ing pathways, and central-member structures.● iondens calculates ion densities.● edyn performs an essential dynamics analysis.● rot_rel calculates the rotational relaxation time for solvent

molecules.● ener calculates any energy for a system.● espmap calculates the vacuum electrostatic potential on a grid

Figure 7. Using the interaction class.

Figure 6. Specialized code generation using templates.


around selected molecules of a given configuration from thepartial charges in the topology.

Miscellaneous

● frameout converts trajectories into other formats or extractssnapshots from trajectories.

● inbox puts the solute into the center of the box.● atominfo prints (topological) information on specified at-

oms.● shake_analysis analyses a specified configuration.● cmt_list lists atoms within a specified distance from a given

atom.

Examples of Application

Local-Elevation Simulation of Glucose

The technique of local-elevation (LE) MD has been developed toenhance the searching of the configurational space of a moleculeby progressively elevating the local potential energy of the con-figurations that are visited during an MD trajectory.135 The totalpotential energy function consists of two terms: the standardphysical terms Vphys(r(t)), and the local-elevation term VLE(r(t),t), which depends also explicitly on the time t. When the molecule

Figure 8. Glucose molecule with atom numbering.

Figure 9. Time course of the six torsional dihedral angles of the glucose ring as obtained fromstandard MD (a) and local-elevation (LE-) MD (b). The LE weight factor was E�

le � 2 kJ mol�1

and the four torsional angles C(1)–C(2)–C(3)–C(4), C(2)–C(3)–C(4)–C(5), C(3)–C(4)–C(5)–O(5), and C(5)–O(5)–C(1)–C(2) were chosen as LE degrees of freedom.


is trapped in a low-energy basin of the potential energy hypersur-face, the LE algorithm gradually elevates the bottom of this basinusing additional Gaussian-shaped energy functions, which willeventually force the molecular system out of the basin into aneighboring basin of the energy hypersurface. In this way, theenergy surface is much more efficiently sampled than using stan-dard MD. For low-dimensional systems LE-MD will lead to a flatpotential energy as soon as all parts of the LE configuration spacehave been visited.135 If

Vphys�r�t�� VLE�r�t�, t� (120)

is flat after a (long) time tl, by construction of the local-elevationpotential energy term, VLE(r, tl) represents the negative of thefree-energy surface or potential of mean force for the LE degreesof freedom of the molecule.

LE-MD was already implemented in GROMOS96.3,4 Here, weillustrate its sampling efficiency using as an example the confor-mational sampling of a glucose molecule (Fig. 8) solvated in SPCwater.136 The time course of the six exocyclic torsional dihedralangles of the sugar ring are shown for a standard MD and forLE-MD simulation in Figure 9. In the standard MD at 300 K and1 atm, no conformational transitions are observed on a 1-ns timescale, while the LE-MD simulation with an energy weight factor

that is raised by 2 kJ mol�1 every time a configuration is revisitedalready leads to a first conformational transition after 200 ps. After500 ps many transitions are observed, indicating that the potentialenergy surface eq. 120 is becoming flat, the free-energy surfacecan then be obtained in the form of �VLE(r, t � 1000 ps).

Replica-Exchange Simulation of Butane

Five hundred butane molecules, all in trans-configuration, havebeen simulated at 273 K. The force constant of the torsional anglehas been increased by a factor 3. The time to reach the equilibriumstate of gauche and trans butane can be determined by monitoringthe width of the torsional-angle distribution. The potential energybarrier between the trans and the gauche configuration is too high(18 kJ/mol) to overcome at 273 K. REMD is applied to increasethe sampling of configurational space at 273 K. To this effect, 11replicas of the system with changed torsional-angle force constants(scaled by 1.0, 0.9, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, and 0.4,respectively) were simulated simultaneously. Every 0.5 ps anexchange between two neighboring replicas was attempted. Figure10 shows the path in � space for all replicas. The overall exchangeprobability during the simulation was 0.25. The time series of theRMSD from the average of the torsional angle of all butanemolecules is depicted in Figure 11. The larger the RMSD, the more

Figure 10. REMD of liquid butane, starting from an all trans configuration of the torsional angle. Thepath in � space for the 11 replicas (starting at � values 0.0, 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55,and 0.6) is shown. Exchanges were attempted every 0.5 ps, in total 250. The overall exchange probabilitywas 0.25. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]


torsional angle transitions from trans to gauche have happened.Through replica exchange, also the simulation running at � � 0contains butane molecules in a gauche conformation, unlike thesimulation carried out without replica exchange. Note that theRMSD is dependent on the width of the valleys, so it is dependenton the force constant. This, in turn, means that the equilibriumvalue of the RMSD is different for the different replicas.

Coarse-Grained Simulation of Alkanes

Coarse-grained (CG) models allow for much more efficient sam-pling of the molecular configurational space than atomic-level(AL) models (at the expense of loss of atomic detail). Yet a CGmodel should be able to reproduce the properties of an AL model,assuming that the latter is correct. To illustrate this requirement forCG models, some properties (conformational distributions andconfigurational entropies) of n-alkanes in the liquid phase137 havebeen compared. The main results are summarized here in thecontext of hexadecane, where the AL model was the standard

GROMOS 45A3 force field129 and the CG model the one dis-cussed before.116 For the AL model 128 hexadecane moleculeswere simulated at 323 K and 1 atm in a periodic box over 25 ns.For the CG model 512 molecules were simulated over 1000 nsunder the same conditions. In the AL model, a hexadecane mol-ecule consists of a linear chain of 16 united atoms. In the CGmodel, four united atoms are represented by one bead, so thathexadecane consists of four beads. To compare AL configurationsof united atoms with CG configurations of beads, the AL trajec-tories were mapped to the CG level by considering only the centersof mass of the four united atoms that represent one bead at the CGlevel. This mapping of the atomic level onto the coarse-grainedlevel is indicated by the symbol MAP.

Figure 12 shows the distribution of the values of the twopseudobond angles and the one pseudotorsional angle of the hexa-decane molecules at the CG level for the MAP (gray) and CG(black) trajectories. The difference in torsional-angle distributioncan be explained from the absence of torsional potential energyterms in the CG model.116 Table 10 shows the configurational

Figure 11. Simulation of liquid butane, starting from an all trans configuration of the torsional angle. Thetime series of the root-mean-square deviation (RMSD) from the average torsional angle is shown. Thebold black line depicts the RMSD in the standard MD simulation. No broadening of the distribution isvisible. The bold red line denotes the RMSD of the replica at �i � 0.0 (corresponding to the standard MDsimulation). Clearly, the relaxation towards the equilibrium state is much faster using the replica-exchangemethod than in the standard MD simulation. The other lines denote the other replicas (at 0.0 � �i � 1.0,many of them reaching their equilibrium state already after about 50 ps. [Color figure can be viewed inthe online issue, which is available at www.interscience.wiley.com.]


entropies of the four united atom fragments of the hexadecanemolecules at the atomic level (AL) and at the CG level (MAP),together with those obtained from the CG simulations (CG). At theCG level the configurational entropies of MAP and CG modelsagree very well, to within 2 J mol�1 K�1.

These data illustrate that the CG model116 is able to reproducethe properties of the GROMOS AL model rather well.

One-Step Perturbation Calculations on the Free Energy ofLigand Binding to the Estrogen Receptor

The one-step approach to calculate relative free energies of com-plexation or ligand binding is particularly efficient when manystructurally not too different ligands are to be considered. Previ-ously,124 the Gibbs free energy of binding of 17 polychlorinatedbiphenyls to the estrogen receptor were calculated from two MDsimulations of an unphysical reference compound, one whenbound to the protein and one free in solution. Here, the efficiencyof the one-step technique is illustrated by calculating more than1500 binding free energies from the two simulations.

Figure 13 shows the biphenyl ligand with the nine atoms thatare made soft atoms in the unphysical reference state. At these ninesoft sites, five different real substituents (H, F, Cl, Br, and CH3)can be placed, yielding 59 � 1 � 1,953,124 relative bindingenergies for the ligands. Here, we calculated free energies for allpossible polyfluorinated, polychlorinated, and polybrominated li-gands (in total 3 � 29 � 1 � 1535 relative free energies) from onesimulation of the free ligand in water and one bound to theestrogen receptor. For every class of substituted biphenyl ligands,the three best binding structures are depicted in Figure 13. (Wenote that the binding affinity of the polybrominated biphenylsmight be underestimated by the choice of the reference state: thesoft-core atoms chosen have smaller van der Waals radii than theBromine atoms.) This application illustrates the efficiency of theone-step perturbation technique for screening purposes in drugdesign.

Other Applications

GROMOS can be used for molecular modelling of any type ofmolecular system. Below, a number of applications are mentioned,which, for convenience, have mainly been taken from our ownmore recent work.

The structural stability of proteins,138–144 peptides,145–152 sug-ars,126,153 and DNA154,155 as function of their composition, chainlengths, and solvent environment or temperature and pressure canbe studied. Solvation, both in pure solvents and in mixtures, can beinvestigated in atomic detail.156–159 Motional properties, NMRcoupling constants, and dielectric relaxation times can be ana-lyzed.139,160–162 3J-coupling constants, NOE’s and NMR orderparameters and CD spectra can be compared to experimentalvalues.163–167 GROMOS can also be used for structure refinement

Table 10. Configurational Entropy (in J K�1 mol�1) of the Four (A–D)Hexadecane Fragments That Correspond to the Four Beads of theCoarse-Grained (CG) Model for Hexadecane in the Liquid Phase.

Fragment AL MAP CG

A 211 133 131B 209 111 110C 209 111 110D 211 133 131

AL: atomic-level entropies; MAP: fragment entropies from the AL trajec-tories; CG: bead entropies from the CG trajectories.

Figure 12. Bond-angle (A–B–C, B–C–D) and torsional dihedral angle(A–B–C–D) distributions at the coarse-grained level. Gray: A–D arecenters of mass of fragments consisting of four united atoms asobtained from AL trajectories. Black: A–D are beads of the CG modelobtained from CG simulations.


of biomolecules based on NMR data.168–170 Molecular host–guestcomplexes can be studied in terms of structural properties and freeenergy and entropy of binding.123,124,127,171–173 Polypeptide(un)folding equilibria can be simulated in atomic detail.174–179

Biochemical reactions can be mimicked in QM/MM simulations,in which interfaces to quantum chemistry software have to beused.180–182

A variety of types of molecules have been simulated: proteins,DNA, RNA, saccharides, lipids, and a range of solvents: water,DMSO, methanol, chloroform, carbontetrachloride, acetonitrile,and mixtures of these and other cosolvents such as urea.183–186

Also, membranes and micelles have been simulated using GRO-MOS.187–192

Conclusions

The GROMOS software for biomolecular simulation has beenextended with new functionality and extended analysis possibili-ties and put partially into C��, which makes extension of func-tionalities easier. GROMOS05 comes with the latest thermody-

namically calibrated GROMOS force field parameter sets 45A3/4and 53A5/6, which are suitable for a broad range of molecularsystems. The source code of GROMOS is obtainable for a nominalfee (http://www.igc.ethz.eh/gromos), and should allow both meth-odological investigations and structural, dynamical, and energeticexplorations of biomolecular systems, which may lead to an en-hanced understanding of the properties of such systems.

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